Arguesian Identities in Invariant Theory

Advances in Mathematics  AI1566 advances in mathematics 122, 148 (1996) article no. 0056

Arguesian Identities in Invariant Theory Michael Hawrylycz Computer Research and Applications Group, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 Received June 24, 1995

Having been motivated by an example of Doubilet, Rota, and Stein [Stud. Appl. Math. 56 (1976), 185216], we present a technique for constructing geometric identities in a GrassmannCayley algebra. Each identity represents a projective invariant closely related to the Theorem of Desargues in the plane and its generalizations to higher dimensional projective space. The construction employs certain combinatorial properties of matchings in bipartite graphs. We also prove a dimension independence result for Arguesian identities, thereby connecting the identities with lattice theory.  1996 Academic Press, Inc.

1. INTRODUCTION Few chapters of twentieth century mathematics have had as strong an influence on the rest of mathematics, and yet have suffered as much from misunderstanding and lack of recognition, as invariant theory. In this brief introduction we should like to recall the main lines of the development, together with the temporary eclipse, of invariant theory in this century, starting with the pioneering work of Alfred Young and Issai Schur, up to the extraordinary resurgence of the subject in the present day. At the heart of the misunderstanding of the program of invariant theory that was formulated by the great geometers of the the past century (mathematicians such as Boole, Clebsch, Gordan, Hermite, Jordan, MacMahon, Cayley, Sylvester, Grace, Capelli, Grassmannn, and the early Hilbert) lie two completely different and at times conflicting trends, both of which had the unfortunate effect of obfuscating the program and purpose of the great masters. On the one hand, the outstanding success of the development of the theory of group representations had the effect of deflecting interest away from the geometric problems that lie at the heart of the subject, and to shift the spotlight onto the newly born subject of abstract algebra. On the other hand, the sweeping results of Hilbert, beginning with his basis 1 0001-870896 18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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theorem, had an opposite effect from the one intended by their discoverer. Instead of stimulating interest in classical invariant theory, they were soon seen as the ideal tool for the rigorization of algebraic geometry. Thus, shortly after the turn of the century, the few remaining stalwarts of echt invariant theory, mathematicians such as Alfred Young, Turnbull, Edge, Grace, Aitken, Study, and Weitzenbock, felt outflanked by representation theorists on one side and algebraic geometers on the other. For these and other reasons the very expression invariant theory came to be associated with passe and outmoded mathematics. The rebirth of invariant theory, together with a clearer understanding of its forgotten program, came slowly starting in the early fifties. Perhaps the turning point can be marked by the publication of the survey Invariant Theory, old and new by Dieudonne and Carrell, which was in fact published as one of the first papers in the journal Advances in Mathematics. In this paper, the emphasis can still be recognized to be on the representation theory of the symmetric and general linear groups, and invariant theory is presented as a series of curious appendages to representation theory. The geometric program underlying nineteenth century invariant theory is hardly mentioned, except for some remarks on Gram's theorem on plane algebraic curves. In essence, the rebirth of geometric invariant theory in its pristine version can be traced to the papers: Doubilet et al. [6], and Barnabei et al. [1], the first of which appeared in the seventies, and the second in 1985. The key idea of classical invariant theory, the idea that was totally neglected for almost a century is quite simple. It begins with the evident remark that the facts of geometry, when expressed in terms of algebra, become dependent upon the choice of a coordinate system. Yet a statement about space can be viewed as a geometric fact only on the condition that such a fact express a property of space which is independent of the choice of a coordinate system. The tendering of geometric facts in algebraic language according to the principles of Cartesian geometry has the unfortunate byproduct of making these facts dependent upon a coordinate system. The idea of an invariant came out of this realization, and the beginnings of invariant theory were motivated by a classification of expressions in Cartesian coordinates that remained invariant under changes of coordinates, and whose vanishing therefore represented some geometric property. A central idea of invariant theory is that a notation has to be devised which is itself independent of the choice of a coordinate system, and which dispenses form the use of coordinates from the start. Thus, the program of classical invariant theory is that of developing a coordinate-free notation for the expression of the facts of geometry, and a systematic translation table of the facts of geometry expresssed in synthetic form into algebraic

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expressions in such a coordinate-free notation. We may briefly compare this program with a strikingly similar program which took place in the development of algebraic geometry this century. The development of the theory of commutative rings led to an understanding of algebraic varieties, and later schemes, whereby the interplay of geometry and algebra is almost entirely subsumed in the interplay between the elements of a commutative ring and the structure of its ideals or modules. In this elegant program, geometric ideas were inevitably viewed as little more than a fanciful language in which to express algebraic facts. The program was successful in more than one way, and for present purposes we stress its success in doing away with coordinates altogether in the study of algebraic varieties. What, then, is the program of classical invariant theory, and how does it compare to the de-coordinatization via commutative rings that has been so successful in algebraic geometry? Again, the answer to this question is simple in retrospect. The emphasis of classical invariant theory lies not in the description of geometric objects, such as algebraic varieties by coordinate-free devices, but in the discovery of a variety of new operations that can be defined among geometric entities in a coordinate-free manner. Therein lies the difference between the two fields, and the statement of the program of classical invariant theory. Only when the notion of operation is fully brought into light can the relevance of the Hilbert finiteness theorems be realized. While the finiteness of generation of invariants in principle gives a guarantee of security, such a guarantee becomes shaky when it is realized that such finiteness is not characteristic-free, as Nagata was first to show. If the spotlight is given to invariant operations rather than invariant expressions, then invariants are to be sought as polynomials, in the sense of universal algebra, definable in terms of the basic operations and syzygies as identities holding among these polynomials. The technical emphasis thus shifts from the commutative algebraic to the combinatorial domain. Furthermore, whereas a statement of finiteness in a charateristicfree context appeared hopeless in the Hilbert-Nagata formulation, such a program is reborn when viewed in terms of operations. It is not only possible but, from a universal algebraic point of view, likely that a characteristic-free finiteness theorem for invariant operations will some day be clearly stated. Motivated by these considerations, in the present work we deal with operations definable in terms of vectors and covectors ( =tensors of steps 1 and n&1 in the exterior algebra of an n-dimensional vector space.) It has been shown by Barnabei et al. [1], that when all is said and done, all invariant operations holding among vectors and covectors can be expressed in terms of two basic operations which are the algebraic rendering of the join and meet of subspaces of a vector space. The existence of these operations goes all the way back to Grassmann, under the name of progressive

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and regressive product. It took however sometime to realize that the algebraic structure required for a correct and projectively invariant definition of a system in which both of these operations can be simultaneously manipulated is not simply an exterior algebra, but an exterior algebra in which an n-linear skew-symmetric scalar valued form is distinguished, which following the language of Hopf algebra, is called the integral. This seemingly insignificant twist makes a great deal of difference. Peano was first to have realized it in three dimensions, and Doubilet et al. [6] in an arbitrary number of dimensions. An exterior algebra together with a distinguished tensor of step n allows two invariant operations: the join, which is the ordinary exterior or wedge product, and which roughly corresponds to the join of vector subspaces, and the meet, which similarly corresponds to the intersections of vector subspaces. The rigorous definition of an algebraic system in which both join and meet are defined is, in our opinion, a notable step forward in the program of invariant theory. First, it allows a restatement of the fundamental theorems of classical invariant theory in terms of these operations. These theorems state that every invariant, as well as all invariant operations, of vectors and covectors, is a polynomial in joins and meets. Second, it shifts the emphasis from the classification of invariants to another fascinating problem: the problem of expressing the facts of projective geometry in terms of identities holding among decomposable skew-symmetric tensors. The topic of the present work is precisely the discovery of a notable class of identities holding among joins and meets, which remarkably enough turn out to correspond in a striking and unexpected way to classical theorems of projective geometry and their generalizations to higher dimensions. Theorems such as Desargues, Pappus, Bricard, Fontene and sundry other jewels of classical synthetic geometry are revealed, as if by magic, to be expressable as simple and elegant identities holding among joins and meets of extensors. These identities yield as a byproduct a host of other theorems in arbitrary dimensions. Our starting point is the analysis of a basic identity that expresses Desargues' theorem in terms of joins and meets, as is found in [6]. Our analysis reveals that the heart of the identity lies in certain multilinearity properties of expressions in joins and meets of vectors and covectors which are unexpected. We observe that identites holding among polynomials in joins and meets that have geometric sigificance are obtained by taking two polynomials, say P and Q, in joins, meets and brackets (the bracket being the distinguished n-linear skewsymmetric form), the first of which is linear in the vector variables but not in the covector variables, and the second of which is linear in the covector, but not in the vector variables. Under certain simple combinatorial conditions, which we derive in full below, P and Q will lead to an identity, when multiplied by suitable powers of the bracket. Interestingly, multiplication

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by the bracket does not change the geometric significance of the identity, but on the contrary inserts genericity conditions which in the synthetic version of the equivalent geometric facts must be stated as verbal provisos. The resulting set of identities can be systematically interpreted as theorems relating to incidences of subspaces in projective space. A large number of new theorems follow from simple geometric interpretation of the identities, most of which would be challenging to prove in classical geometric terms, either synthetically or by using homogeneous coordinates. To illustrate, consider the following theorem due to Fontene in three-dimensional projective space. Theorem [Fontene ] Let a, b, c, d and a$, b$c$, d $ be the vertices of two tetrahedra in projective three space. Intersect the lines aa$, bb$, cc$ and dd $ with the faces b$c$d $, a$c$d $, a$b$d $ and a$b$c$ of tetrahedron a$, b$, c$, d $. Then these four points are coplanar if and only if the four planes formed by joining the lines bcd & b$c$d $, acd & a$c$d $, abd & a$b$d $ and abc & a$b$c$, which are the intersection of opposite face planes of the tetrahedra, to the points a, b, c, d, all pass through a common point. This theorem is proved in a GrassmannGayley algebra of dimension 4 by the identity. [a, b, c, d] 3 (aa$ 7 b$c$d $) 6 (bb$ 7 a$c$d$) 6 (cc$ 7 a$b$d $) 6 (dd$ 7 a$b$c$) =[a$, b$, c$, d $](bcd 7 b$c$d $) 6 a) 7 ((acd 7 a$c$d $) 6 b) 7 ((abd 7 a$b$d $) 6 c) 7 ((abc 7 a$b$c$) 6 d ). The proof and interpretation of the Fontene's identity is given below. Ideally, our identities would be proven in the context of superalgebras [10, 14]. To this date, however, the meet as an operation in supersymmetric algebra has not been rigorously defined, and such attempts have led to contradictory results, or results which are difficult to interpret. A recent announcement by Brini [3] indicates that the theory of Capelli operators and Lie superalgebras may provide the required setting. It may be recalled that another program proposed this century for the algebraic rendering of expressions involving joins and meets of subspaces of vector spaces; is the program of lattice theory. Modular lattices, and more recently after Haiman [11], and Finberg et al. [7], linear lattices, (that is lattices of commuting equivalence relations), were seen as an analog of Boolean algebra that might be suitable for such a purpose. Although the possibility of such a program has not been established, it is likely that a

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host of invariant facts about subspaces of vector spaces should be expressible as identities holding in linear lattices and thus in modular lattices as well. It was evident from the start of lattice theory with Dedekind, that reasoning with the modular law, or with the proof theory recently developed by Haiman or Finberg et al., is far from transparent. Nonetheless, the question naturally arises whether any of the identities we derive in the present work for the GrassmannCayley algebra can be translated into identities holding in linear lattices. Although we do not complete this program in the present work, we obtain what we believe is an important step. The step consists in stating an identity holding in the Grassmann Gayley algebra in sufficiently general form, as to allow the replacement of vectorcovector variables by variables corresponding to decomposable skew-symmetric tensors of arbitrary steps. The condition under which such a substitution can be made is perhaps the deepest result of the present work; at any rate it is a result that has no precedents in the previous literature on exterior algebra. Once such a substitution property is established for a given identity, it comes natural to conjecture that a closely related identity, in which algebraic joins and meets are replaced by latticial joins and meets, will hold in linear or modular lattices. With this conjecture we close the present introduction and summarize some of the essential results Doubilet et al. and Baranabei et al..

2. THE GRASSMANNCAYLEY ALGEBRA A Peano space is a vector space equipped with the additional structure provided by a form with values in a field. The definition of a Peano space, the exterior algebra of a Peano space, and the basic properties of these structures were first developed by Doubilet et al. [6] and later Barnabei et al. [1]. We state only some of their results for completeness and the reader is referred to these papers for a more complete treatment. Let K be an arbitrary field, and let V be a vector space of dimension n over K, which will remain fixed throughout. Given vector space V, let S(V) denote the free associative algebra on V, and G(V) its exterior algebra. The product in the exterior algebra of a Peano space is called the join, and is denoted by the symbol . We note that this usage differs from the ordinary usage where exterior multiplication is denoted as the wedge product . Let ,: S(V)  G(V) denote the canonical projection of S(V) onto G(V). If x 1 , x 2 , ..., x k is a word in S(V) with x 1 , x 2 , ..., x k # V, for k>0, we denote its image under , by ,(x 1 , x 2 , ..., x k )=x 1 6 x 2 6 } } } 6 x k and provided ,(x 1 , x 2 , ..., x k ){0 the element is called the extensor x 1 , x 2 , ..., x k of step k.

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Proposition 2.1. Let A be a subspace of V of dimension k>0; if [x 1 , x 2 , ..., x k ] and [ y , y 2 , ..., y k ] are two bases of A then x 1 6 x 2 6 } } } 6 x k =Cy 1 6 y 2 6 } } } 6 y k for some non-zero scalar C. By Proposition 1.2 every non-trivial subspace of V is uniquely represented, modulo a non-zero scalar, by a non-zero extensor and vice-versa. The zero subspace is represented by scalars. We say that the extensor x 1 } } } x k is associated to the subspace generated by the vectors of V corresponding to [x 1 } } } x k ]. We also remark that the join a 1 6 } } } 6 a k is non-zero if and only if the set of associated vectors is a linearly independent set. The following proposition is fundamental. Proposition 2.2. Let A, B be two subspaces of V with associated extensors F and G respectively. Then 1. F 6 G=0 if and only if A & B{[0]. 2. If A & B=[0], then the extensor F 6 G is the extensor associated to the subspace generated by A _ B. A second operation in the exterior algebra of a vector space is the meet. A precursor to this operation was originally recognized by Hermann Grasssman [9] in his famous Ausdehnungslehre. Grassmann's intention was to develop a calculus for the geometry of linear varieties, and the equivalent of the meet was called the regressive product, unfortunately denoted by the same notation as the join or wedge product. While this operation was later used by authors such as Whitehead [20] and Fordor [8], the realization that the exterior algebra of a Peano space, with its two operations of join  and meet , is the natural structure for the study of projective invariant theory under the special linear group was not made until [6]. Given an extensor A=a 1 6 a 2 6 } } } 6 a k and an ordered r-tuple of non-negative integers h 1 , h 2 , ..., h r such that h 1 +h 2 + } } } +h r =k, a split of type (h 1 , h 2 , ..., h r ) of the representation A=a 1 6 a 2 6 } } } 6 a k is an ordered r-tuple of extensors (A 1 , A 2 , ..., A r ) such that 1. A i =A if h i =0 and A i =a i 6 a i2 6 } } } 6 a ih i if h i {0, 2. A i 6 A j {0, if i{j, 3. A 1 6 A 2 6 } } } 6 A r = \A. In what follows we shall denote by S(a 1 , a 2 , ..., a k ; h 1 , h 2 , ..., h r ) the finite set of all splits of type (h 1 , h 2 , ..., h r ) of the extensor A relative to the representation A=a 1 6 } } } 6 a k .

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The definition of the meet of two extensors is based on the following fundamental property of Peano spaces. Proposition 2.3. Let a 1 , a 2 , ..., a k and b 1 , b 2 , ..., b p be vectors of a Peano space V of dimension n with k+pn. If A=a 1 6 a 6 } } } 6 a k and B=b 1 6 b 2 6 } } } 6 b p then the following identity holds: sgn(A 1 , A 2 )[A 1 , B] A 2

: (A1, A2 ) # S(A; n&p, k+p&n)

=

:

sgn(B 1 , B 2 )[A, B 2 ] B 1 .

(B1, B2 ) # S(B; k+p&n, n&k)

We may now define the meet of two extensors A and B. Given extensors A=a 1 6 a 6 } } } 6 a k and B=b 1 6 b 2 6 } } } 6 b p with k, p1, we define the binary operation 7 by setting: 1. A 7 B=0, if k+p
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Proposition 2.6. Let a 1 , a 2 , ..., a k be vectors and X 1 , X 2 , ..., X s covectors, with ks. Set A=a 1 a 2 } } } a k , then: A 7 (X 1 7 X 2 7 } } } 7 X s ) =

:

sgn(A 1 , A 2 , ..., A s+1 )

(A1, ..., As+1 ) # S(A;1, ..., 1, k&s)

_[A 1 , X 1 ][A 2 , X 2 ] } } } [A s , X s ]A s+1 . Example 2.7. (a 1 6 a 2 6 a 3 ) 7 (X 1 7 X 2 ) = [a 1 , X 1 ][a 2 , X 2 ] a 3 & [a 1 , X 1 ][a 3 , X 2 ] a 2 & [a 2 , X 1 ][a 1 , X 2 ] a 3 + [a 2 , X 1 ][a 3 , X 2 ] a 1 + [a 3 , X 1 ][a 1 , X 2 ] a 2 &[a 3 , X 1 ][a 2 , X 2 ] a 1 Corollary 2.8. covectors; then

Let a 1 , a 2 , ..., a n be vectors and X 1 , X 2 , ..., X n be

(a 1 6 } } } 6 a n ) 7 (X 1 7 } } } X n )=det([a i , X j ]) i, j=1, 2, ..., n The double bracket of covectors X 1 , X 2 , ..., X n denoted [[X 1 , X 2 , ..., X n ]] is defined to be the scalar X 1 7 X 2 7 } } } 7 X n . We may conclude from the properties of the meet that the double bracket is also non-degenerate and is of step zero, a scalar. Thus, the vector space spanned by the covectors is of dimension n. A set of covectors with non-zero double bracket constitutes a basis of covectors. In this case, a corresponding basis of vectors a1 , a 2 , ..., a n can be found satisfying X i =a 1 } } } a^ i } } } a n . A simple calculation shows that [[X 1 , ..., X n ]]=[a 1 , ..., a n ] n&1. Let V be a Peano space of dimension n over the field K. We say that a linearly ordered basis [a 1 , a 2 , ..., a n ] of V is unimodular whenever [a 1 , a 2 , ..., a n ]=1. If a 1 , a 2 , ..., a n is a unimodular basis, the extensor E=a 1 6 a 2 6 } } } 6 a n is called the integral. The integral is well-defined and does not depend on the choice of unimodular basis. For details and properties of unimodular bases the reader is referred to [1]. The meet operation defines a second exterior algebra structure on the vector space G(V). The duality operator connecting the two is the Hodge Star Operator. Given a linearly ordered basis [a 1 , a 2 , ..., a n ], the associated cobasis of covectors of V is the set of covectors [: 1 , ..., : n ] where : i =[a i , a 1 , ..., a^ i , ..., a n ] &1 a 1 6 } } } 6 a^ i 6 } } } 6 a n .

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Let [a 1 , a 2 , ..., a n ] be a linearly ordered basis of V. The Hodge star operator relative to the basis [a 1 , a 2 , ..., a n ] is defined to be the (unique) linear operator V : G(V)  G(V) such that, for every subset S of [1, 2, ..., n], V 1=E, V a i1 6 } } } 6 a ik =(&1) i1 + } } } +ik &k(k+1)2[a 1 , ..., a n ] &1 a p1 6 } } } a pn&k , where if S=[i 1 , ..., i k ] with i 1
V maps extensors of step k to extensors of step n&k,

(ii) V(x 6 y)=(Vx) 7 (Vy) and V(x 7 y)=(Vx) 6 (V y), for every x, y # G(V), (iii)

V1=E and VE=1,

(iv)

V(Vx)=(&1) k(n&k) x, for every x # G k(V).

Following notation of [1] we shall have need for the notion of the split of an extensor written as the meet of covectors. If A is an extensor and A=X 1 7 } } } 7 X k with X 1 , X 2 , ..., X k covectors, given an ordered s-tuple of non-negative integers k 1 , k 2 , ..., k s such that k 1 + } } } +k s =k, a cosplit of type (k 1 , ..., k s ) of the extensor A=X 1 7 } } } 7 X k is an ordered s-tuple of extensors such that (i)

A i =E if k i =0 and A i =X i1 7 X i2 7 } } } 7 X ik i if k i {0,

(ii)

A i 7 A j {0, if i{j,

(iii)

A 1 7 A 2 7 } } } 7 A r = \A,

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Denote C(X 1 7 } } } 7 X k ; k 1 , ..., k s ) the set of all cosplits of type (k 1 , ..., k s ) of the set of covectors [X 1 , ..., X k ]. There is a dual expansion for covectors whose proof follows easily from Hodge duality. Proposition 2.11. Let A 1 , A 2 , ..., A k and B 1 , B 2 , ..., B p be covectors in a Peano space V of step n with k+pn. Setting A$=A 1 7 A 2 7 } } } 7 A k and B$=B 1 7 B 2 7 } } } 7 B p the the following identity holds: sgn(A$1 , A$2 )[[A$1 , B$]] A$2

: (A$1 , A$2 ) # C(A; n&p, k+p&n)

=

sgn(B$1 , B$2 )[[A$, B$2 ]] B$1 .

:

(1)

(B$1 , B2$ ) # C(B; k+p&n, n&k)

Given A=X 1 7 } } } 7 X k , B=Y 1 7 } } } 7 Y p with k, p1, we define A 6 B=0 if k+ps. Set A=X 1 7 X 2 7 } } } 7 X k . Then A 6 (a 1 6 } } } 6 a s )=

:

sgn(A 1 , ..., A s+1 )

(A 1, ..., A s+1 ) # C(A; 1, ..., k&s)

_[A 1 , a 1 ][A 2 , a 2 ] } } } [A s , a s ] A s+1

3. ARGUESIAN POLYNOMIALS We introduce a class of expressions in a GrassmannCayley algebra called Arguesian polynomials, as each represents a projective invariant closely related to the configuration of Desargues' Theorem in the projective plane. In GC(n) let a=[a 1 , a 2 , ..., a n ] be an n-set of rank 1 vectors and X=[X 1 , X 2 , ..., X n ] be an n-set of rank n&1 covectors. We let lowercase letters denote vectors and uppercase letters denote covectors. The variable set a (respectively X) occurs homogeneously of order l in a GC expression P if each a # a (X # X) occurs l1 times in P. The variable set a (respectively X) occurs multilinearly in P if each a # a (respectively X # X) occurs exactly once in P. Definition 3.1. A type I (resp. type II ) Arguesian polynomial P(a, X) in GC(n) is an expression in join  and meet , on multilinear (resp. homogeneous) variable set a and homogeneous (resp. multilinear) variable set X.

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A type I basic extensor e is an expression of the form a 1 } } } a k 6 X 1 } } } X l for lh. A type II basic extensor has lk and the meet  replacing the join . An Arguesian polynomial is P trivial if P can be written as the product of brackets, each bracket consisting only of vectors or only of covectors. Given QP, Let V(Q) denote the subset (not multiset) of vectors of a occurring in Q and C(Q) the subset of covectors of X occurring in Q. We remark that if Arguesian type III P has order l, a calculation shows that P is necessarily of full-step. An Arguesian polynomial P is proper if every proper subexpression of P has positive rank. The following identity is due to Doubilet et al [6]. Theorem 3.2. (Desargues). The corresponding sides of two coplanar triangles intersect in colinear points if and only if joins of the corresponding vertices are concurrent. As a GC(3) identity, [a, b, c]((a 6 BC) 7 (b 6 AC)) 6 (c 6 AB) =[[A, B, C]](bc 7 A) 6 (ac 7 B) 6 (ab 7 C) Proof.

We prove the equivalent identity, [a, b, c](a 6 BC) 7 (b 6 AC) 7 (c 6 AB) } E =[[A, B, C]](bc 7 A) 6 (ac 7 B) 6 (ab 7 C).

The Arguesian polynomial P in step 3 (a 6 BC) 7 (b 6 AC) 7 (c 6 AB)

(2)

is expanded using Proposition 2.12 to obtain, (B[a, C]&C[a, B]) 7 (A[b, C]&C[b, A]) 7 (A[c, B]&B[c, A])).

(3)

The meet of any two common covectors must vanish, hence by the linearity of meet, (3) becomes &BCA[a, C][b, A][c, B]+CAB[a, B][b, C][c, A]

(4)

Q=(bc 7 A) 6 (ac 7 B) 6 (ab 7 C)

(5)

Also,

may be similarly expanded as ([b, A] c&[c, A] b) 6 ([c, B] C&[c, B] a) 6 ([a, C] b&[b, C] a) =&[b, A][c, B][a, C] cab+[c, A][a, B][b, C] bca

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(6)

ARGUESIAN IDENTITIES IN INVARIANT THEORY

Fig. 1.

13

Desargues' Theorem.

Interchanging the positions of any two vectors (or covectors) changes the sign. Since the extensor abc is of step 3 while ABC is an extensor of step 0, we may cross multiply expressions (4) and (6) by these factors, putting E on the left to balance rank, to obtain the given identity. A somewhat more appealing form is obtained by taking a new basis a$, b$, c$ setting A=b$c$, B=a$c$ and C=a$b$. Hence ABC=[a$, b$, c$] 2; and we obtain after cancellation, [a, b, c][a$, b$, c$](aa$ 7 bb$ 7 cc$) E =(bc 7 b$c$) 6 (ac 7 a$c$) 6 (ab 7 a$b$)

(7)

The identity (7) may now be easily interpreted: Assuming the points a, b, c and a$, b$, c$ are in general position, the left side vanishs, most generally, when the intersection of lines aa$ and bb$ lies on the line cc$, or the three lines are concurrent. Since (7) is an algebraic identity the left side vanishes iff the right side vanishes, which occurs when the line formed by joining points bc & b$c$, ac & a$c$ contains the point ab & a$b$, or the three points are colinear. For a synthetic proof, see [4]. K E

Given Arguesian polynomials P and Q, define P # Q, read P is E-equivalent to Q, if there is r in field K such that the identity P=rQ is valid in a GC algebra, where we allow that either side may be multiplied by the integral E. In the case of Arguesian polynomials E-equivalence incorporates the fact that the scalar brackets [a 1 , ..., a n ], [[X 1 , ..., X n ]] and the overall sign difference of P and Q have no bearing on the geometry. We shall have need to distinguish between the l homogeneous occurrences of the covectors (vectors) of a type I (II) polynomial P, replacing the covector X j # X by distinct X j1 , X j2 , ..., X jl (and similarly for vectors.) The resulting

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polynomial is called the repeated variable representation P*(a, X*) of P, and we shall say that X ji is a repeated covector of label X j . We shall often write X* to denote a generic repeated covector of label X. The expansions of Propositions 2.3, 2.6, 2.11, and 2.12 may be recursively applied to a type I, (or type II) P*(resp. Q*) of order l, as a multilinear polynomial in l } n covectors (vectors), the resulting expansion having no cancellation of terms. This expansion, in which a monomial contains brackets [a, X*], [[X ji , ..., X lm ]], is defined to be the repeated alternative expansion E (P*) of P*, and as every variable of P* is distinct, each monomial of E (P*) occurs with scalar coefficient \1. The expansion E (Q*) is well-defined for subexpressions Q*P*, where it signifies the linear combination of extensors, and brackets [a, X ji ], [[X ji , ..., X lm ]] over the field K. For type I QP, denote [a, X ji ] # E (Q*) to mean the bracket [a, X ji ] occurs amongst the brackets of E (Q*). If R is a vector or covector E (R*)=R*, and if R=S 6 T or R=S 7 T (which we denote as S  T) then E (R*)= E (S*) E (T *). If G(a, X*) (resp. G(a, X)) denotes the exterior algebra generated by vectors a and covectors X*, (resp. X), and I is the ideal of G(a, X*) generated by relation X ij &X il for X ij &X il # X*, then G(a, X*)I$G(a, X) under the canonical projection \: G(a, X*)  G(a, X*)I. It is clear that \ is an algebra homomorphism, and if A*, B* denote elements of G(a, X*), then \(A*  B*)=\(A*)  \(B*), where the join and meet are evaluated in G(a, X*) and G(a, X) respectively. The following canonical expansion [6], shall be used throughout this paper. Proposition 3.3. Any non-trivial non-zero type I Arpuesian polynomial of order l in GC(n) can be written in the form: P=[[X 1 , X 2 , ..., X n ]] l&1 : C _[a 1 , X _(1) ][a 2 , X _(2) ] } } } [a n , X _(n) ]

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_

where X _ is a permutation of the covector set X and C _ is an integer constant depending on _. Proof. If P has step n we may write P=P$ 6 E where E is the integral and P$ has step 0. Therefore assume that P*(a, X*) is type I step 0 and consider the projection of E (P*) under the homomorphism \. By multilinearity of a in P, no monomial M of the projection contains a bracket [a 1 , ..., a n ], unless P is trivial. Then M contains the product of n scalar brackets [a, X], one for each a # a, and brackets [[X 1 , ..., X n ]] whose covectors are precisely X. As X occurs homogeneously in P, the covectors in brackets [a, X] in M must be the entire set X as well. Finally, each bracket [[X 1 , ..., X n ]] may be linearly ordered and factored from the expansion. K

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Proposition 3.4. Any non-trivial non-zero type II Arguesian polynomial Q of order m in GC(n) can be written in the form: Q=[a 1 , a 2 , ..., a n ] m&1 : C _[a _(1) , X 1 ][a _(2) , X 2 ] } } } [a _(n) , X n ] _

where a _ is a permutation of the vector set a and C _ is an integer constant depending on _. Definition 3.5. Given a non-trivial non-zero type I or type II Arguesian polynomial P, the bracket polynomial E (P)=: C _[a 1 , X _(1) ][a 2 , X _(2) ] } } } [a n , X _(n) ]

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_

defined by either Propositions 3.3 or 3.4, is called the Alternative expansion E (P) of P. Definition 3.6. Given an Arguesian polynomial P(a, X), a transversal ? is a bijection ?: a  X such that the monomial [a 1 , X ?(1) ][a 2 , X ?(2) ] } } } [a n , X ?(n) ] occurs with non-zero coefficient C ? in E (P). We shall denote by E (P)| ? the non-zero monomial of E (P) determined by ?. Example 3.7. The map ?: a  A, d  C, b  D, c  B is a transversal of the type I Arguesian (((a 6 AB) 7 C) 6 d) 7 ((b 6 CD) 7 A) 7 (c 6 BD) with corresponding non-zero monomial +[a, A][b, D][c, B] _[b, D]. Given type I Arguesian P and QP, let E (Q) denote the projection of E (Q*) under \ (excluding brackets [[X 1 , ..., X n ]].) The resulting expression we call the partial alternative expansion E (Q) of Q. If Q=R  S, then E (Q)=E(R)  E(S). Let [a, X] # E (P) denote that the bracket with content [a, X] occurs in some monomial of E (P). If [a 1 , ..., a k ] (resp. [X 1 , ..., X l ]) denotes a set of vectors (resp. covectors) contained in the support of the extensors of positive step of linear combination E (Q), (welldefined as step Q>0, for any proper QP), we shall write Q(a 1 , ..., a k ) (resp. Q(X 1 , ..., X l )) to make this explicit. The notations Q*(a 1 , ..., a k ) and are similarly defined by E (Q*). Thus, [a, X*] # E (P*) if Q*(X * 1 , ..., X *) l and only if _R 6  7 SP with a # V(R), X* # C(S*), and R*(a), S*(X*).

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Definition 3.8. A subexpression QP of an Arguesian polynomial of either type is type I (resp. type II ) if E (Q*) is a linear combination (resp. Q*(a 1 , ..., a k ), for a set of covectors [X * Q*(X * 1 , ..., X *), l 1 , ..., X *]X* l (vectors [a 1 , ..., a k ]a). Example 3.9. The type I polynomial P=(a 6 BC) 7 (b 6 AC) 7 (c 6 AB) in repeated representation is P*=(a 6 B 1 C 1 ) 7 (b 6 A 1 C 2 ) 7 (c 6 A 2 B 2 ). Then E (P*)=(B 1[a, C 1 ]&C 1[a, B 1 ]) 7 (A 1[b, C 2 ]&C 2[b, A 1 ]) 7 (A 2[c, B 2 ]&B 2[c, A 2 ]) and expanding by linearity of meet yields the terms, +B 1 A 1 A 2[a, C 1 ][b, C 2 ][c, B 2 ]&B 1 A 1 B 2[a, C 1 ][b, C 2 ][c, A 2 ] &B 1 C 2 A 2[a, C 1 ][b, A 1 ][c, B 2 ]+B 1 C 2 B 2[a, C 1 ][b, A 1 ][c, A 2 ] &C 1 A 1 A 2[a, B 1 ][b, C 2 ][c, B 2 ]+C 1 A 1 B 2[a, B 1 ][b, C 2 ][c, A 2 ] +C 1 C 2 A 2[a, B 1 ][b, A 1 ][a, B 2 ]&C 1 C 2 B 2[a, B 1 ][b, A 1 ][c, A 2 ]. Since the meet of any two covectors of the same letter type is zero, only two of the terms survive in E (P). If Q*=a 6 B 1 C 1 then Q is type I, E (Q*)= B 1[a, C 1 ]&C 1[a, B 1 ], and Q*(B 1 , C 1 ). In studying the transversals of Arguesian polynomials, the following definition is useful. Definition 3.10. A pre-transversal of a type I Arguesian polynomial P*(a, X*) is a map f *: a  X* such that the projection f: a  X is a bijection, and f *: a i  X j* only if [a i , X j*] # E (P*). Given QP, a pre-transversal f * identifies a set of monomials [E (Q*)| f * ] of E (Q*) as follows: M # [E (Q*)| f * ] iff \[a, X*] # M, f *: a  X*, a # V(Q), X* # C(Q*). As C(P)=X for any Arguesian P, an easy induction shows, Proposition 3.11. Given Arguesian P, QP, and a pre-transversal f *, there is at most one monomial of [E (Q*) | f * ] having non-zero projection E (Q) | f under \. If E (Q) | f is non-zero under f *, we denote the unique monomial of E (Q*) defined by Proposition 3.11 as E (Q*) | f * . Its extensor of positive step, if non-empty, is denoted ext(E (Q*) | f * ). If the projection E (Q) | f is non-zero, its extensor is denoted as ext(E (Q) | f ). We may write [a, X] # E (Q) | f to indicate that the bracket [a, X] occurs amongst the brackets of the monomial E (Q) | f . Write X # ext(E (Q) | f ) to mean ext(E (Q) | f ) is the

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meet of covectors one of which is X. Since the vectors of type I P are multilinear, an easy induction establishs: Proposition 3.12. Let P be type I Arguesian, QP a type I subexpression, and let f *, g* be pre-transversals of P* having non-zero projections E (Q) | f , and E (Q) | g . If [a, X] # E (Q) | f  [a, X] # E (Q) | g , then ext(E (Q) | f )=ext(E (Q) | g ), i.e. the extensors of positive step are identical. We shall require a few technical lemmas. Lemma 3.13. Let f * be a pre-transversal of a type I Arguesian P*, QP, with E (Q)| f non-zero, and X # C(Q). Then 1. Let QP be type I. If [a, X] # (resp.  ) E (Q) | f , for some (resp. any) a # V(Q), and X  (resp. # ) ext(E (Q) | f ), then: For any pre-transversal g* with E (Q) | g non-zero, [b, X] # E (Q) | g for some b # V(Q) iff X  ext(E (Q) | g ). 2. Let QP be type I. If [a, X] # (resp.  ) E (Q) | f , for some (resp. any) a # V(Q), and X # (resp.  ) ext(E (Q) | f ), then: For any pre-transversal g* with E (Q) | g non-zero, [b, X] # (resp.  ) E (Q) | g , for some (resp. any) b # V(Q), and X # (resp.  ) ext(E (Q) | g ). 3. Let Q/P be type II. If [a, X] # E (Q) | f , for a # V(Q), then: For every pre-transversal g* of P* with E (Q) | g non-zero, there is b # V(Q) such that [b, X] # E (Q) | g . Proof. We prove cases 1-3 simultaneously by induction QP. A basic extensor Q=e has unique vectors and covectors, and 1 and 3 are clear. For 2, X # ext(E (e) | f ) iff [a, X]  E (e) | f . Let Q=R 7 S for type I R, S. Case 1. Suppose [a, X] # E (Q) | f and X  ext(E (Q) | f ). Then without loss of generality [a, X] # E (R) | f , [b, X]  E (S) | f , for any b # V(S). If [b, X] # E (Q) | g , and in particular [b, X] # E (R) | g , then if X # ext(E (R) | g ), by 2 applied to R, X # ext(E (R)| f ), a contradiction. As [b, X]  E (S)| f , X  ext(E (S) | f ), 2 applied to S yields [b, X]  E (S) | g for any g. If conversely, [b, X]  E (Q) | g then [b, X]  E (R) | g , so [a, X] # E (Q) | f and X  ext(E (R) | f ) imply, by part 1 applied to R, X # ext(E (R) | g ). Then by 2 applied to S, X  ext(E (S) | g ), so X # ext(E (Q) | g ). The proof of the parenthesized case follows identically. Case 2. This case is proved analogously.

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Case 3. If [a, X] # E (R)| f , for a # V(R), the Lemma holds by induction. Suppose [a, X] # E (S)| f , for a # V(S). Then if X # ext(E (S)| f ), as Q is type II _b # ext(E (R)| f ), with [b, X] # E (Q)| f , a contradiction. Hence X  ext(E (S)| f ), and S satisfies 1. Then \g* with E (S)| g non-zero [a, X]  E (S)| g implies X # ext(E (S)| g ) and again [b, X] # E (Q)| g for b # V(R). If finally, [a, X] # E (Q)| f for a # ext(E (R)| f ), X # ext(E (R)| f ) then again S satisfies 1, and the result follows. K The following Lemma is fundamental to Arguesian polynomials. The Lemma is false when the assumption of multilinearity is dropped. Lemma 3.14. Let P be a non-zero type I Arguesian polynomial with transversal ?. Then for any QP, there is a unique monomial E (Q*)| ?* of E (Q*) having non-zero projection \

 E (Q)| ? . E (Q*)| ?* w Proof. If Q is the join of vectors or meet of covectors, then E (Q*)=Q*, E (Q)=Q, and the result is trivial. The Lemma is also clear when Q is a type I (II) basic extensor. Let Q=R 7 S with R, S type I. For any pre-transversal ?* with E (Q)| ? non-zero, [a, X] # E (Q)| ? implies [a, X] # E (R)| ? or [a, X] # E (S)| ? . By Proposition 3.12, the brackets [[a, X]] of E (R)| ? and E (S)| ? uniquely determine ext(E (R)| ? ), and ext(E (S)| ? ). Hence E (Q)| ? factors uniquely as \  E (R)| ? , E (R)| ? 7 E (S)| ? . By induction there are unique E (R*)| ?* w \  E (S)| ? and as \ is a homomorphism of algebras E (S*)| ?* w def

\

E (Q*)| ?* # E (R*)| ?* 7 E (S*)| ?* w  E (R)| ? 7 E (S)| ? =E (Q)| ? is the required monomial of E (Q*). Let Q=R 6 S, for type I R, S. The argument is identical and as C(P)=X, a single monomial E (Q*)| ?* of E (R*)| ?* 6 E (S*)| ?* survives in the projection under \. Let Q=R 6 S for type II R, type I S. Let g* be a pre-transversal of P* with E (Q)| g non-zero. If a # V(S) then [a, X] # E (S)| g , for some X # C(S), and by Proposition 3.12, ext(E (S)| g ) is determined by the set of brackets [a, X] # E (S)| g . By Lemma 3.13 (part 3), given any pre-transversal g* with E (Q)| g non-zero, the covectors X # C(R) satisfying [a, X] # E (R)| g , determine a set C/C(R) such that [a, X] # E (R)| ? for some a # V(R) for any ? iff X # C. Thus we conclude, for every [a, X] # E (Q)| ? , 1. [a, X] # E (S)| ? iff a # V(S), 2. [a, X] # E (R)| ? iff X # C, 3. [a, X] # E (Q)| ?

with a # V(R)

X # C(S)"C otherwise.

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Then ext(E (R)| ? ), ext(E (S)| ? ) are uniquely determined and E (R)| ? 6 E (S)| ? is the unique factorization of E (Q)| ? with corresponding unique \ \  E (R)| ? , and E (R*) ?* w  E (S)| ? . As the covectors of E (R*) ?* w ext(E (S)| ? ) are distinct, the brackets of third type above determine a unique map ?: a  X, a # V(R), X # C(S)"C, and a unique monomial E (Q*) ?* of E (R*) ?* 6 E (S*) ?* having projection E (Q)| ? under \. The proof is dual for Q=R 7 S with R type II, S type I. K Corollary 3.15. Given an Arguesian polynomial P and transversal ?, the coefficient C ? of E (P)| ? is always \1. Corollary 3.15 motivates the following definition. Definition 3.16. Given an Arguesian polynomial P with transversal ? the coefficient C ? of E (P)| ? is called the sign of ? and denoted sgn(E (P)| ? ). The following Lemmas, whose proofs are elementary, will be necessary for the proof of Theorem 4.1. Proposition 3.17. (Grassmann Condition for Arguesian Polynomials). If f * is a pre-transversal of type I Arguesian polynomial P* but E (P)| f =0 then either, 1. There exists type I R 7 SP with R, S type I, E (R)| f , E (S)| f nonzero, and X j # X such that X j # ext(E (R)| f ) and X j # ext(E (S)| f ),

or

2. There exists type I R 6 SP with R, S type I, E (R)| f , E (S)| f nonzero, and X j # X such that X j  ext(E (R)| f ) and X j  ext(E (S)| f ), Example 3.18. The bijection f: a  F, b  E, c  A, d  B, e  C, f  D corresponds to a pre-transversal of P=((a 6 ADF ) 7 (b 6 ACE)) 6 ((c 6 AEF ) 7 (d 6 BCD)) 6((e 6 BCE) 7 ( f 6 BDF )) yet f is not a transversal, as if R=(a 6 ADF ) 7 (b 6 ACE) then E (R)| f = [a, F][b, E] ADAC=0. Lemma 3.19. Let QP is a type I subexpression of type I Arguesian P, be pre-transversals with E (Q)| f1 , E (Q)| f2 non-zero. If and let f * 1 , f* 2

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X ji # ext(E (Q*)| f *i ), i=1, 2, for X j1 {X j2 # X* of common label X j , then for every pre-transversal f * with E (Q)| f non-zero, there is a # V(Q) such that [a, X] # E (Q)| f . Lemma 3.20. Let P be a non-zero type I Arguesian polynomial of step n homogeneous of order l with vector set a and repeated covector set X*=[X ji , j=1, ..., n, i=1, ..., l]. Let ?, _ be two transversals of P, with corresponding pre-transversals ?*, _*. Then for any a # a, if [a, X ji ] # E (P*)| ?* and [a, X ji $ ] # E (P*)| _* then i=i $. We conclude this section with a definition. Definition 3.21. Given an Arguesian polynomial P(a, X) define the associated graph B p =(a _ X, E) to be the bipartite multigraph on vertex sets a and X, having edge (a, X) # E if [a, X*] # E (P*) for some X* # X* of label X.

4. ARGUESIAN IDENTITIES We present a general construction for identities between Arguesian polynomials. In general, the Grassmann condition Proposition 3.17 makes the construction of Arguesian identities quite complicated, however Theorem 4.1 gives a construction fundamental to all Arguesian identities. Particularly interesting are GC algebra identities for the higher Arguesian lattice identities and an n-dimensional generalization of Bricard's Theorem [2]. Theorem 4.1 (Main Theorem on Arguesian Identities). Let B=(a _ X, E) be a simple bipartite graph on vertex sets a, X having a perfect matching. For a # a, form type I basic extensors e a =a 6 [X j ] where X # [X j ] if (a, X) # E. Similarly, for X # X, form type II basic extensors f X =[a i ] 7 X. Let P (resp. Q) be a type I (II ) Arguesian polynomial formed recursively from [e a ] _ X (resp. f X _ a) using the (dual ) rules 1. Given type I T, with C(T )=[Y i ] multilinear in X and e a = a 6 [X j ] with [Y i ][X j ] set T$=a 6 (T 7 ([X j ]"[Y i ])). 2. Given type I S and T, form S 7 T. E

Then if P and Q both have order 2, P # Q. If P and Q have order l, m3 with P= li=1 Q i , Q= mj=1 P j and each Q i , P j is multilinear in both vectors E and covectors, then P # Q.

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A large class of geometric identities in higher-dimensional projective space are consequences of Theorem 4.1. We first illustrate the theorem with examples. Corollary 4.2. In a four-dimensional projective space, the intersection of the solid abd $e$ and the line a$b$c$ when joined with the point c yields a plane P 1 . The two planes da$c$ and b$d $e$ when intersected and joined to the point e give a line l 1 . The planes cde and a$b$c$ when joined with the line ab yield a plane. Intersect this plane with the solid a$c$d$e$ to obtain a line l 2 . Intersect the solid abce with the plane b$d $e$ to obtain another line l 3 . Then the plane P 1 and line l 1 contain a common point iff the lines l 2 , l 3 and the point d lie on a common hyperplane. Proof.

In GC(5), let P=((ab 6 ABC) 7 DE) 6 c) 7 ((d 6 BDE) 7 AC) 6 e).

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The basic extensors f j obtained from associated B p are abce 7 A, abcde 7 B, abce 7 C, cde 7 D, and cde 7 E. Applying dual rule 1, form cde 7 DE from cde 7 D and cde 7 E. The resulting extensor may be combined using dual rule 1 with abcde 7 B to form ((cde 7 DE) 6 ab) 7 B). Similarly, combine abce 7 A, abce 7 C to form abce 7 AC. Finally, by two applications of the dual to rule 2, join these expressions with vector d to form: . Q=(((cde 7 DE) 6 ab) 6 (abcd 7 AC) 6 d

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Then by Theorem 4.1 part a) we have (((ab 6 ABC) 7 DE) 6 c) 7 ((d 6 BDE) 7 AC) 6 e) E

# (((cde 7 DE) 6 ab) 7 B) 6 (abce 7 AC) 6 d. K

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The following theorem is attributed to Raoul Bricard [2]. Corollary 4.3. (Bricard). Let a, b, c and a$, b$, c$ be two triangles in the projective plane. Form the lines aa$, bb$, and cc$ joining respective vertices. Then these lines intersect the opposite edges b$c$, a$c$, and a$b$ in colinear points if and only if the join of the points bc & b$c$, ac & a$c$ and ab & a$b$ to the opposite vertices a, b, and c form three concurrent lines (see Fig. 2.). Proof. In a GC(3), let a, b, c be vectors and A, B, C be covectors. Then the identity, [a, b, c] 2(a 6 BC) 7 A) 6 ((b 6 AC) 7 B) 6 ((c 6 AB) 7 C) =[[A, B, C]] 2(bc 7 A) 6 a) 7 ((ac 7 B) 6 b) 6 ((ab 7 C) 6 c)

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Fig. 2.

Bricard's Theorem.

is valid by Theorem 4.1. Upon specialization A=b$c$, B=a$c$, C=a$b$ one obtains, [a, b, c] 2(aa$ 7 b$c$) 6 (bb$ 7 a$c$) 6 (cc$ 7 a$b$) E

# [a$, b$, c$]((bc 7 b$c$) 6 a) 7 ((ac 7 a$c$) 6 b) 6 ((ab 7 a$b$) 6 c) (13) The left side of (13) vanishes when the points a, b, c are non-colinear and the join of the points aa$ 7 b$c$, bb$ 7 a$c$, forming a line in the projective plane, when joined to the point cc$ 7 ab, do not span the plane, i.e. the three points are colinear. Interpreting the right side of (13), the two lines (bc 7 b$c$) 6 a and (ac 7 a$c$) 6 b intersect in a point in the plane. The right side vanishes when the join of this point and the line corresponding

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to the extensor (ab 7 a$b$) 6 c do not span the plane, or the three lines are concurrent. Corollary 4.4 (Fontene ). Let a, b, c, d and a$, b$c$, d $ be the vertices of two tetrahedra in projective three space. Intersect the lines aa$, bb$, cc$ and dd$ with the faces b$c$d $, a$c$d $, a$b$d $ and a$b$c$ of tetrahedron a$, b$, c$, d $. Then these four points are coplanar if and only if the four planes formed by joining the lines bcd & b$c$d $, acd & a$c$d $, abd & a$b$d $, and abc & a$b$c$, which are the intersection of opposite face planes of the tetrahedra, to the points a, b, c, d, all pass through a common point. Proof. In GC(4), let a, b, c, d be points and A, B, C, D be planes. Then the identity, [a, b, c, d] 3(a 6 BCD) 7 A) 6 ((b 6 ACD) 7 B) 6 ((c 6 ABD) 7 C) 6 ((d 6 ABC) 7 D) E

# [[A, B, C, D]] 3((bcd 7 A) 6 a) 7 ((acd 7 B) 6 b) 7 ((abd 7 C) 6 c) 7 ((abc 7 D) 6 d ) is valid. Whereupon specialization of A=b$c$d $, B=a$c$d $, C=a$b$d $, D=a$b$c$ one obtains, [a, b, c, d] 3(aa$ 7 b$c$d$) 6 (bb$ 7 a$c$d $) 6 (cc$ 7 a$b$d $) 6 (dd $ 7 a$b$c$) E

# [a$, b$, c$, d $]((bcd 7 b$c$d $) 6 a) 7 ((acd 7 a$c$d $) 6 b) 7 ((abd 7 a$b$d$) 6 c) 7 ((abc 7 a$b$c$) 6 d ).

K

The following theorem appears in [12] and is not difficult to give its geometric interpretation. Corollary 4.5 (N-dimensional Bricard). Let a 1 , ..., a n be vectors and X 1 , ..., X n be covectors in a GrassmanCayley algebra of step n. Then the following identity is valid : n

[a 1 , a 2 , ..., a n ] n&1  ((a i 6 X 1 X 2 } } } X i } } } X n ) 7 X i ) i=1 n

=[[X 1 , X 2 , ..., X n ]] n&1  ((a 1 a 2 } } } a^ i } } } a n 7 X i ) 6 a i ) i=1

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24

MICHAEL HAWRYLYCZ

or upon substituting X i =a$1 a$2 } } } a^$i } } } , a$n for 1in, n

[a 1 , a 2 , ..., a n ] n&1  (a i a$i 7 a$1 a$2 } } } a^$i } } } a$n ) i=1 n

=[a 1 , a 2 , ..., a n ]  ((a 1 a 2 } } } a^ i } } } a n 7 a$1 a$2 } } } a^$i } } } a$n ) 6 a i ). i=1

Corollary 4.6. In four-dimensional projective space, let a, b, c, d, e and a$, b$, c$, d $, e$ be two sets of points. Then the points determined by the intersection of five pairs of planes aba$ 7 cde, bcb$ & ade, cdc$ & abe, ded $ & abc and aee$ & bcd all lie in a common three-dimensional hyperplane if and only if the five three-dimensional solids determined by joining the lines a$e$, a$b$, b$c$, c$d $, d$e$ respectively to the lines bcde & b$c$d $, acde & c$d$e$, abde & a$d $e$, abce & a$b$e$ and abcd & a$b$c$ all contain a common point. Proof.

In a GC(5) we have

[a, b, c, d, e] 4((a 6 CDE) 7 AB) 6 ((b 6 ADE) 7 BC) 6 ((c 6 ABE) 7 CD) 6 ((d 6 ABC) 7 DE) 6 ((e 6 BCD) 6 AE) E

# [[A, B, C, D, E]] 4(ae 6 (A 7 bcd)) 7 (ab 6 (B 7 cde)) 7 (bc 6 (C 7 ade)) 7 (cd 6 (D 7 abe)) 7 (de 6 (E 7 abc))

K

We may obtain a GC algebra identity for the Arguesian lattice law. Corollary 4.7 (Arguesian Law). In a Grassmann-Cayley algebra of step n let the vector set a be partitioned into three sets [a 1 , a 2 , ..., a k1 ], [b 1 , b 2 , ..., b k2 ], and [c 1 , c 2 , ..., c k1 ] of sizes k 1 , k 2 and k 3 respectively, with k 1 +k 2 +k 3 =n and set a (k1 ) =a 1 a 2 } } } a k1 , b (k2 ) =b 1 b 2 } } } b k2 and c (k3 ) = c1 c 2 } } } c k3 . Similarly, partition the covectors set X into sets [X1 , X 2 , ..., X l1 ], [Y 1 , Y 2 , ..., Y l2 ], and [Z 1 , Z 2 , ..., Z l3 ], with l 1 +l 2 +l 3 =n, setting X (l1 ) =X 1 X 2 } } } X l1 , Y (l2) =Y 1 Y 2 } } } Y l2 and Z (l3 ) =Z 1 Z 2 } } } Z l3 . Then the following is an identity in GC(n) provided l1 +l 2 >k 3 , l 2 +l 3 >k 1 , l 1 +l 3 >k 2 , [a (k1 ), b (k2 ), c (k3 ) ](a (k1 ) 6 Y (l2 )Z (l3 ) ) 7 (b (k2 ) 6 X (l1 )Z (l3 ) ) 7 (c (k3 ) 6 X (l1 )Y (l2 ) ) E

# [[X (l1 ), Y (l2 ), Z (l3 ) ]](a (k1 )b (k2 ) 7 Z (l 3 ) ) 6 (a (k1 )c (k3 ) 7 Y (l2 ) ) 6 (b (k2 )c (k3 ) 7 X (l1 ) )

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ARGUESIAN IDENTITIES IN INVARIANT THEORY

25

Proof. It suffices to remark that l 1 +l 2 >k 3 , l 1 +l 2 +l 3 =n, k 1 +k 2 + k 3 =n implies l 3
If k 1 +k 2 +k 3 =n and l 1 +l 2 +l 3 =n

[a (k1 ), b (k2 ), c (k3 ) ][a$ (l1 ), b$ (l2 ), c$ (l3 ) ](a (k1 )a$ (l1 ) 7 b (k2 )b$ (l2 ) 7 c (k3 )c$ (l3 ) ) E

# (b (k2 )c (k3 ) 7 b$ (l2 )c$ (l3 ) ) 6 (a (k1 )c (k3 ) 7 a$ (l1 )c$ (l3 ) ) 6 (a (k1 )b (k2 ) 7 a$ (l1 )b$ (l2 ) ) Corollary 4.9. Let a (2), b (2) be lines and c be a point in projective four space, and a$ a point and b$ (2), c$ (2) lines. Then the plane formed by joining a (2)a$, the solid formed by joining lines b(2)b$ (2), and the plane formed by joining cc$ (2) contain a common point, if and only if the line formed by intersecting the plane b (2)c with the solid b$ (2)c$ (2), the point formed by intersecting the planes a (2)c, a$c$ (2), and the line formed by intersecting the solid a (2)b (2) with the plane a$b$ (2), all lie in a common solid. The Arguesian identities given by Corollary 4.8 are direct consequences of the Arguesian lattice identity. Any lattice equality is equivalent to a lattice inequality, and it can be shown [11] that the Arguesian law may be written c 7 ([a 6 a$) 7 (b 6 b$)] 6 c$) a 6 ([((a 6 b) 7 (a$ 6 b$)) 6 ((b 6 c) 7 (b$ 6 c$))] 7 (a$ 6 c$)) (14) where the operations join  and wedge  are lattice theoretic join and meet. The equivalence of identity 4.8 to the Arguesian lattice identity in the case where the flats corresponding to a (k1 ), b (k2 ), c (k3 ) and a$ (l1 ), b$ (l2 ), c$ (l3 ) are in general position is easily seen. Identity (14) was shown by Haiman [11] to hold in all linear lattices, lattices representable by commuting equivalence relations of a set, and is therefore valid in the lattice of subspaces of a projective space. Assuming that the flats corresponding to a, b, c have the zero element as their meet, the lattice elements a 6 a$, b 6 b$ represent the subspace of V spanned by a, a$ and b, b$. Intersecting these two subspaces and joining the resulting flat with the flat c$, then meeting

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MICHAEL HAWRYLYCZ

with c, the result gives c or the zero element depending on whether the subspace configuration contained a common point. It is the zero element precisely when the left side of the identity 4.8 vanishes, and in this case the subspaces are centrally perspective. On the right side of (14), the clause in square brackets is the subspace containing (a 6 b) 7 (a$ 6 b$) and (b 6 c) 7 (b$ 6 c$) which assuming general position of the subspaces represented by the extensors, corresponds to the flat (a (k1 )b (k2 ) 7 a$ (l1 )b$ (l2 ) ) 6 (b (k2 )c (k3 ) 7 b$ (l2 )c$ (l3 ) ). Meeting the subspace a$ 6 c$ then joining with a we obtain a subspace passing through c only when the given term lies on a common hyperplane with (a (k1 )c (k3 ) 7 a$ (l1 )c$ (l3 ) ). We conclude that the Arguesian law, in the case of subspaces in general position, is realizable as a set of GrassmannCayley algebra identities. Corollary 4.10 (M th Higher Order Arguesian Law). In GC(n) let the vectors set a be partitioned into m+3 sets [a i1 , a i2 , ..., a i k i ] of sizes k i for i a i j . Similarly partition the covector set X into 1im+3. Set a i(ki ) = kj=1 i sets [X i1 , X i2 , ..., X l i ], setting x (li ) = lj=1 X ij . Then provided l i +l i+1 >k i and k i +k i+1 >l i+1 , for i=1, ..., m (the order being such that m+1=1), the following identity is valid: m+3 (ki ) (li+1 ) 2) m+3 ) , ..., a (k 6 X (li i ) X i+1 ) [a 1(k1 ) , a (k 2 m+3 ]  (a i i=1 E

m+3

(lm+3 ) (li+1 ) i+1 ) ]]  (a i(ki ) a (k # [[X 1(l1 ) , X 2(l2 ) , ..., X m+3 i+1 7 X i+1 ) i=1

Corollary 4.11. Let a, b, c, d and a$, b$, c$, d $ be two sets of points in three-dimensional projective space, and consider the two sets of lines ab, bc, cd, ad and a$b$, b$c$, c$d $, a$d $. Then the four planes ac$d $, ba$d $, ca$b$, db$c$ all pass through a common point if and only if the four points formed by intersecting the lines ab, bc, cd, ad with the corresponding planes a$c$d $, a$b$d $, a$b$c$, b$c$d $ all line on a common plane. Proof.

The identity (see Fig. 3)

[a, b, c, d](a 6 AB) 7 (b 6 BC) 7 (c 6 CD) 7 (d 6 AD) E

# [[A, B, C, D]](ab 7 B) 6 (bc 7 C) 6 (cd 7 D) 6 (ad 7 A) is valid in GC(4). Substituting A=b$c$d$, B=a$c$d $, C=a$b$d $, D=a$b$c$ we obtain [a, b, c, d][a$, b$, c$, d $](ac$d $ 7 ba$d $ 7 ca$b$ 7 db$c$) =(ab 7 a$c$d $) 6 (bc 7 a$b$d $) 6 (cd 7 a$b$c$) 6 (ad 7 b$c$d $)

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K

27

ARGUESIAN IDENTITIES IN INVARIANT THEORY

Figure 3

To understand the higher Arguesian identities we proceed as follows. The Nth higher Arguesian law as given by Haiman [11] may be written, given alphabets of letters a 1 , a 2 , ..., a n , and b 1 , b 2 , ..., b n as n&1

\_  (a 6 b )& 6 b + a 6 \_  ((a 6 a

an 7

i

i

n

i=1

n&1

1

i

i=1

i+1

) 7 (b i 6 b i+1 )

+& 7 (b 6 b )) 1

n

(15)

Proposition 4.12 (Rota). Any lattice identity PQ is equivalent to one in which every variable appears exactly once on each side. By applying Proposition 4.12 the Nth higher Arguesian law may be written in the following self-dual form. N th Higher Order Arguesian Law. Let a 1 , ..., a n , a$1 , ..., a$n and b 1 , ..., b n , b$1 , ..., b$n be alphabets. Then the following identity holds as a linear lattice identity:

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28

MICHAEL HAWRYLYCZ n&1

\_  ((a 7 a$) 6 (b 7 b$))& 6 (b 7 b$ )+ a 6 \_  ((a$ 6 a ) 7 (b$ 6 b ))& 7 (b 6 b$ )+

a$n 7

i

i

i

i

n

n

i=1

n&1

1

i

i+1

i

i+1

1

n

(16)

i=1

In identity (16) let A 1 , A 2 , ..., A n , B 1 , B 2 , ..., B n be new variables and substitute b i =A i , b$i =A i+1 with b$n =A 1 and a i =a$i =B i . Then (16) becomes the following linear lattice identity, after application of the lattice rules B i 7 B i =B i , A i 6 A i =A i and the commutativity of lattice theoretic join and meet, n&1

\_  (B 6 (A 7 A )+& 6 (A 6 A )  B 6 \ \_  ((B 6 B ) 7 A )& 7 A ++

Bn 7

i

i

i+1

1

n

i=1

n&1

1

i

i+1

i+1

1

(17)

i=1

The left hand side of this identity is zero when the subspace B n 6 (A 1 7 A n ) has some point in common with the bracketed term on the left hand side of (16). Meeting both sides of (17) with B n the left hand side remains invariant while the right hand side vanishes when A 1 6 (B 1 7 B n ) lies on a common hyperplane with the bracketed term on that same side. These are precisely the conditions making the left and right hand sides of (4.10) vanish. The identity (16) has a natural geometric interpretation. If a 1 b 1 , ..., a n+1 b n+1 are n+1 concurrent sets of lines in projective n-space. Then the n+1 points, whose intersection must exist, a 1 a 2 & b 1 b 2 , a 2 a 3 & b 2 b 3 , ..., a 1 a n & b 1 b n all lie on a common hyperplane. Haiman [11] has shown that the (N+3)-rd higher Arguesian law is a strictly stronger lattice identity than N th order law. It is conjectured that the (N+1)-rst is strictly stronger than the N th.

5. PROOF OF THE MAIN THEOREM We verify that Arguesian P and Q constructed according to rules 1,2 have the same transversals occurring with equal or opposite sign uniformly. By construction, X # C(e a )  a # V( f X ), and therefore _X* # X* iff _a* # a* of label a such that of label X such that [a, X*] # E (e *) a Suppose e a =a 6 [X j ], and [a, X l1 ] # E (e a*) for [a*, X] # E ( f * X ). X l1 # X* of label X l # [X j ]. Let T be type I with C(T )=[Y i ][X j ], and

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ARGUESIAN IDENTITIES IN INVARIANT THEORY

29

apply rule 1. Forming T $=a 6 (T 7 ([X j ]"[Y i ]), if X l # [X j ]"[Y i ] then [a, X l1 ] # E (T $*). If X l # [Y i ], then X l # C(T ). As C(T*) contains no repeated labels of X, and T contains no type II subexpressions other than the join of vectors, _X l2 # X* of label X l such that T*(X l2 ) and then [a, X l2 ] # E (T $*). As [a, X*] # E ((S  T )*) iff [a, X*] # E (S*) or [a, X*] # E (T *). We conclude that [a, X l1 ] # E (e a*) for X l1 of label X l iff # E (P*). Therefore, for each pre-trans_X l* of the same label with [a, X *] l versal of P there corresponds a pre-transversal of Q with identical bijection ?: a  X, and conversely. Suppose P has order 2. By Proposition 3.17, if P is zero under pretransversal ?*, there is T = R 7 S  P with X j # ext(E (R)| ? ) and X j # ext(E (S)| ? ). Let X j1 # ext(E (R*)| ?* ) and X j2 # ext(E (S*)| ?* ). Then there is no a # V(R) _ V(S) with ?*: a  X j1 or ?*: a  X j2 . Since ?*: b  X ji for some i # [1, 2] and b # a, TS$, with R$ S$P and b # V(R$)"V(T ). But then C(R 7 S) has repeated X j1 , X j2 of label X j so R$ 6 S$ is not formed. Let P= li=1 Q i , l3. Grassmann condition 1 does not apply to any Q i as C(Q i )=X multilinearly. If SQ i is type I with P of any order satisfying the hypotheses of Theorem 4.1, then an easy induction shows X* # C(s*) implies S*(X*). If ?* is a pre-transversal of P with ?*: a  X*, X* # X*, with X* of label X, the covector of label X appears in l&1 distinct ext(E (Q i )| ? ), 1il, and the join E (P)| ? = li=1 E (Q i )| ? is non-zero. Hence P and Q have the same transversals, and by Corollary 3.15 all transversals occur with coefficient \1. Let ? and _ be transversals of P. At least one exists by hypothesis. When no confusion results we shall identify ? with its corresponding transversal ?&1 of Q. We construct a sequence of transversals ?=? 0 , ? 1 , ..., ? s =_ in which sgn(E (P)| ? )_sgn(E (P) | ?i+1 )=sgn(E (Q)| ?i )_sgn(E (Q)| ?i+1 ) from which it follows that, sgn(E (P)| ? )_sgn(E (P)| _ )=sgn(E (Q)| ? )_sgn(E (Q)| _ )

(18)

E

and P # Q. If ? and _ are transversals of P then by Lemma 3.20, if ?*: a  X jl and _*: a  X jm for X jl , X jm # X* of label X j then l=m. Thus ? and _ induce a permutation of X defined as \: ?(a i )  _(a i ), and it suffices to verify (18) for the case of \ a cycle. Set V( \)=[a i # a | ?(i){_(i)], C( p)=[X # X | ? : a  X, a # V(\)], and denote | \| as the cardinality of V( \). Lemma 5.1. Let ?, _ be two transversals of a type I Arguesian P(a, X) in GC(n) and suppose that the induced permutation \ of X is a cycle. Then there is a sequence of transversals ?=? 0 , ? 1 , ..., ? m&1 , ?$m , ? m+1 , ..., ? s =_

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MICHAEL HAWRYLYCZ

such that the permutation induced by ? i , ? i+1 is a transposition for all i{m, and if i=m, \ m induced by ?$m , ? m+1 is a transposition or a cycle satisfying: For a # V(\ m ), if [a, X j*] # E (P*)| ?*m+1 for X j* # X* of label X j , then there . does not exist b # V( \ m ), such that [b, X j*] # E (P*)| ?$* m Proof. If ? and _ differ by a transposition, or ?=_, the Lemma is trivial. Let C t denote the permutation induced by the pair ? t , _, 0ts. # E (P*)| _* , [b, X *] # E (P*)| ?*t for a, b # V(C t ), and 0ts, If [a, X *] j j let ? * t : a  X l* , ? t* : b  X j* , _*: a  X j* , and _*: b  X r* with X j* , X l* , X r* # X* of labels X j , X l , X r # X. Then by the construction of the theorem either Case 1. _R 6 SP with R type II, S type I and linear combinations R*(a, b), S*(X * j , X l* , X r*). Case 2. _T=R 6 S/P with R type II, S type I, R*(b), S*(X * j , X* i , X r*), and T/S$ with S$ 6 R$P, S$ type I, R$ type II, # E ((R$ 6 S$)*). R$*(a). Therefore, S$*(X * j , X* i , X r*) and [a, X *] l Case 3. _T=R 6 S/P with R type II, S type I, R*(b), S*(X * j , X *), r and T/S$ with S$ 6 R$P and S$ type I, R$ type II R$*(a), S$*(X * j , X* i , X r*) and X l*  C(S*). In case 1 or 2, set ?* t+1(c)=? t*(c) if c  [a, b] and ?* i+1 : a  X * j , ?* i+1 : b  X l* . Then as ? i+1(a)=_(a) and ? i+1(b)=? t(a), no new violations of cases 1-3 occur. Since ?* i+1 is a pre-transversal, ? i+1 is a transversal, and if C t is a cycle of length l then C t+1 is a cycle of length l&1. Given ?=? 0 , eliminate the m-occurrences of case 1 or 2, by the above reassignment, to form ?=? 0 , ? 1 , ..., ? m&1 , ?$m . Every violation of _*: a  X * j , ?$* m: b  X* j for X j* # X*, a, b # V(C m ) is therefore of the form of case 3. Let X i* # X* and consider a maximal length sequence ?$* m : ai  X * i , _*: a i  X* i+1 , for 1ik. By construction of P for every i there exists R i 6 S i P with R i (a i ), S i* (X i* , X* i+1 ). We claim the sequence [R i 6 S i , 1ik] satisfies R i+1 6 S i+1 /S i , where the inclusion is proper, or else case 1 or 2 applies. As R i 6 S i P, , we must have either R i* (a i ), S i* (X i* , X*i+1 ), and [a i+1 , X* i+1 ] # E (P*)| ?$* m a i+1  V(R i 6 S i ), a i+1 # V(R i ) or a i+1 # V(S i ). In the first case, there is type I S$ with R i 6 S i /S$, R$ 6 S$P, and R$*(a i+1 ), S$*(X * i , X* i+1 ). Then the case 2 transposition ?* m+1 : a i  X* i+1 , ?* m+1 : a i+1  X i* applies, a contradiction. Similarly, if a i+1 # V(R i ) case 1 applies. As S i is type I, then R i+1 6 S i+1 S i . The inclusion must be proper, for else R i 6 (R i+1 6 S i+1 )P and then (R i 6 R i+1 ) 6 S i+1 P and case 1 applies. Thus for 1ik, R i+1 6 S i+1 /S i , and we may write,

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ARGUESIAN IDENTITIES IN INVARIANT THEORY

R 1(a 1 )

S 1*(X * 1 , ..., X* k , X* k+1 )

R 2(a 2 )

S *(X * 2 2 , ..., X* k , X* k+1 )

}}} R k(a k )

31

}}} S* k(X* k , X* k+1 )

Form ? m+1 as follows: For each maximal sequence of the above type set ?m+1 : a 1  X*k+1 , leaving fixed ?* m+1 : a  X i* , 2ik. Further, if a # V(C m ) such that there does not exist b # V(C m ) with ?$* m(b)=_*(a), set (a)=_*(a). As ?$ , _ are bijections, ? is a well-defined transversal, ?* m+1 m m+1 and the cycle \ m induced by ?$m and ? m+1 has | \ m | < |C m |. The transversal _ may be recovered from ? m+1 by a sequence of case 2 transpositions applied to each maximal sequence. As R *(a 1 1 ), * , ..., X * , X* ), and R* (a ), S* (X* , ..., X* S *(X 1 1 k k+1 k&i+1 k&i+1 k&i+1 k&i+1 k+1 ), (a)=?* (a) for a{a , a for 1ik, define ? m+i+1 as ?* m+i+1 m+i 1 k&i+1 , : a  X* , ?* : a  X* . Thus at step i, and ?* m+i+1 1 k&i+1 m+i+1 k&i+1 k&i+2 ? m+i+1(a k&i+1 )=_(a k&i+1 ), as |C m | = | \| &m, and there is a bijection between the set of transversals, [? i |im+2], and the set a, b # V(C m ) some X* of label X] [X # C(C m ) | _*(a)=? $*(b)=X*, m

K

Lemma 5.2. If the cycle \ m induced by ?$m , ? m+1 satisfies the property of Lemma 5.1 then, sgn(E (P) | ?m$ )_sgn(E (P) | ? m+1 )=

&1

{&1

| \ m | &1

order P=2 order P3

Proof. If | \ m | =2, then by Lemma 7.1 the Lemma is true. Let RP in which R contains no join of type I subexpressions, and P has any order. Given TR, let ext(E (T )| ?$m )=X 1 } } } X p , and ext(E (T )| ?m+1 )=Y 1 } } } Y p . We claim by induction, that ext(E (T )| ?m+1 ) can be reordered, without affecting sgn(E (P)| ?$m )_sgn(E (P)| ?m+1 ), so that ext(E (T )| ?$m ) and reordered sgn(E (T )| ?m+1 ) satisfy; * For all j=1, ..., p, either 1) X *=Y j j or, 2) if X * j {Y j* then X j {Y j , and X j {Y k for j{k and there is a # V(T ) & V(\ m ) such that ?*m+1 : a  X j* , $* : a  Y * and ? m j . The result is trivially valid if T=eR is a type I basic extensor, as 1 is satisfied. Also if T=T 1 7 T 2 with T 1 , T 2 type I, since by induction T 1 , T 2 satisfy 1 or 2, the result is valid. Thus let T=T 1 6 T 2 with T 1 type II and T 2 type I. We may write ext(E(T 1 )| ? m$ ) 6 ext(E(T 2 )| ? $m )=a 1 } } } a k 6 X 1 } } } X l , ext(E(T 1 )| ? m+1 ) 6 ext(E(T 2 )| ? m+1 )=a 1 } } } a k 6 Y 1 } } } Y l , and

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MICHAEL HAWRYLYCZ

E (T 1 6 T 2 )| ?m+1 =Y 1 } } } Y l "[Y ?m+1(1) , ..., Y ?m+1(k) ] _sgn(E (T 1 )| ?m+1 )_sgn(E (T 2 )| ?m+1 )

(19) (20)

_sgn(Y 1 } } } Y l "[Y ?m+1(1) , ..., Y ?m+1(k) ], Y ?m +1(1) , ..., Y ?m+1(k)

(21)

and similarly for E (T 1 6 T 2 )| ?m$ . Assuming the claim Let a # [a 1 , ..., a k ] and ? $* m : a  X* i , ?* m +1 : a  Y *. j holds by induction for E (T 2 )| ?$m and E (T 2 )| ?m+1 , ther are cases. * Case 1. If i=j and X *=Y i i . Then a  V( \ m ), X i  ext(E (T)| ?$m ), X j  ext(E (T)| ?m+1 ). The case i{j, X i =Y j , does not occur, as then X i {Y i , (? m+1 is non-zero), contradicting the inductive hypothesis. * and X *=Y * then X j # ext(E (T)| ?$m ), Case 2. If i{j, X *=Y i j j j , Y i # ext(E (T)| ?m+1 ), and ?* m+1(a)=X j*=Y j* , ?$* m(a)=Y i*=X i* . $*(a), Case 3 If i=j but X i {Y i , then by induction, X i*=?* m+1(b)=? m $*(b)=?* and Y i*=? m m+1(a), for b # V( \ m ) & V(T 2 ). Then | \ m | =2, and Lemma 5.2 is valid. Case 4. If i{j, X i {Y i , or X j {Y j , then assuming the latter, there is ? $*(b)=Y * ? $*(b)= b # V(\ m ) & V(T 2 ), with ?* m+1(b)=X j* , m j . Then m a # V(T"T 2 ), and a{b, contradicting Lemma 5.1, unless ?* m +1(a), | \ m | =2. If | \ m | 3, the elements of [a 1 , ..., a k ] & V(\ m ) are assigned by ?$m , ? m+1 as in Case 2. By hypothesis there does not exist a i , aj # [a 1 , a 2 , ..., a k ] & V(\ m ) with ?* m+1(a j )=?* m(a i ), the position of ?$* m+1(a i ) in X 1* } } } X* p is distinct from the position of ?* m+1(a i ) in Y * 1 } } } Y* p , and both are distinct from either of the positions of ?* m+1(a j ) and ?$* m(a j ) for j{i. Thus, the covectors of (19) occuring in both (19) and (20) may be simultaneously reordered, without affecting sgn(E (P)| ?$m )_sgn(E (P)| ?m+1 ), to satisfy the claim. Furthermore, it is easy to see that sgn(E (T 1 6 T 2 )| ?$m ) _sgn(X 1 , ..., X l ) differs from the reordered sgn(E (T 1 6 T 2 )| ?m+1 )_ sgn(Y 1 , ..., Y l ) by the parity of |[a 1 , ..., a k ] & V(\ m )|. Suppose P= li=1 Q i , l3. Then sgn(E (P)| ?m+1 )_sgn(E (P)| ?$m ) is given by (&1) | \m | , times the parity change of sgn( li=1 ext(E (Q i )| ?m$ )), from reordered sgn( li=1 ext(E (Q i )| ?m+1 ). For i=1, ..., l let the j th covector of ext(E (Q i )| ?$m ) and reordered ext(E (Q i )| ?m+1 ) be denoted X i, j and Y i, j . Let P=[(i, j)|X i, j {Y i, j ]. As ? m+1 is a bijection and C(Q i )=X, it is easy to see that [X i, j ] (i, j) # P =[Y i, j ] (i, j) # P , and the map \$: X i, j  Y i, j , for (i, j) # P, defines a cycle of length | \ m |. Applying Lemma 5.5 (the assumption of unimodularity is irrelevant by E-equivalence), the

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ARGUESIAN IDENTITIES IN INVARIANT THEORY

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sign of  li=1 ext(E (Q i )| ?$m ) and reordered  li=1 ext(E (Q i )| ?m+1 differ, and therefore sgn(E (P)| ?$m )_sgn(E (P)| ?m+1 )=(&1) | \m| &1 The case of Arguesian order 2 is similar. K We now complete the proof of Theorem 4.1. Given ?, _, transversals of P, ? &1, _ &1 induce a permutation \ &1 of a in the obvious way. If \ induces a cycle CP , then the set of edges [(a i , ?(a i )), (a i , _(a i ))], for a i # V(CP ) form a cycle in B. The same cycle of B may be equivalently defined as [(X j , ? &1(X j )), (X j , _ &1(X j ))], for X j # C(CP ), and \ &1 induces a cycle CQ in a of length |CP | = |CQ |. Let ?, _ be transversals of P, inducing a cycle CP , and let ?=? 0 , ? 1 , ..., ? m&1 , ?$m , ? m+1 , ..., ? s =_

(22)

be the sequence given by Lemma 5.1. Consider (22) as a sequence of transversals of Q. As \ &1 , induced by ? i , ? i+1 , in Q, 0im&1, and i m+1is, is a transposition, we may apply Lemma 7.1 to obtain, sgn(E (P)| ? )_sgn(E (P)| ?$ )=sgn(E (Q)| ? )_sgn(E (Q)| ?$m ), sgn(E (P)| ? m+1 )_sgn(E (P)| _ )=sgn(E (Q)| ?m+1 )_sgn(E (Q)| _ ). &1 If \ &1 m is a transposition, then Theorem 4.1 is true. Hence assume \ m is &1 a cycle |C$Q | 3. By Lemma 5.5, |C$Q | < |CP |. If \ m satisfies the dual to Lemma 5.5, then Theorem 4.1 is true. Otherwise, apply dual Lemma 5.1 to transversals ?$m , ? m+1 of Q, substituting the resulting sequence ?$m =` 0 , ..., `$q , ` q+1 , ..., ` r =? m+1 , (with r |C$Q | ), for ?$m , ? m+1 in (22). If C"P denotes the cycle \ q induced by `$q , ` q+1 then by Lemma 5.5, |C"P | < |C$Q | < |CP |. We may iterate this procedure obtaining a finite refinement

?=# 0 , # 1 , ..., #$p , # p+1 , ..., # t =_

(23)

with t |CP |, where # i , # i+1 , i{p differ by a transposition, and as |CP | is finite, #$p , # p+1 differs by a transposition or a cycle simultaneously satisfying Lemma 5.1 in P and dual Lemma 5.1 in Q. Therefore sgn(E (P)| #$p )_sgn(E (P)| #p+1 )=sgn(E (Q)| #$p )_sgn(E (Q)| #p+1 ) and Theorem 4.1 follows.

K

The connection between Arguesian polynomials of order two, and those of higher order is implicit in the proof of Theorem 4.1. Corollary 5.3 itself gives a large class of geometric identities.

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Corollary 5.3. If Arguesian P has order 2, and Q has order l3, where P and Q are constructed from the same bipartite graph B using E Theorem 4.1, then P # Q if and only if the permutation \ induced by any pair of transversals ? and _ is even. The following definition and Lemma complete the proof of Theorem 4.1. Following Forder [8] we define, Definition 5.4. Given the ordered basis a=[a 1 , ..., a n ] of a GC algebra of step n, and an extensor e=a 1 } } } a l of step 0ln, define the supplement |e to be the unique extensor [a 1 , ..., a l , a$1 , ..., a$n&l ], a$1 , ..., a$n&l where [a 1 , ..., a$n&l ]/a is the set of basis elements linearly independent from [a 1 , ..., a l ]. In the case where the basis a is unimodular we may write |a 1 } } } a l =a l+1 } } } a n . Lemma 5.5. Let X 1 , ..., X n be a unimodular basis of covectors and let Ai , i=1, ..., l be ordered sets of distinct covectors C(Ai )/X, with l

 7 Ai =[[X 1 , ..., X n ]] l&1. i=1

Let [Cj ] j=1, ..., d /X for 2dn be a set of distinct covectors each occupying a fixed position of some Ai . Let \ be a permutation of [C j ] which is a cycle, having the property that if C j # C(Ai ), for some i # [1, ..., l] then \(C j )  C(Ai ). Let Bi , i=1, ..., l be the ordered sets of covectors formed by setting Bi =Ai except in a position of Ai occupied by an element C j which is replaced by \(C j ). Then,  li=1 7 Bi = \[[X 1 , ..., X m ]] l&1 and sgn

\

l

+

 7 Ai = &sgn i=1

\

l

 7 Bi i=1

+

Proof. It is not difficult to show that the join  li=1 7 Bi is non-zero and that it suffices to prove the Lemma for the case where each Ai contains at most one element of [C j ] j=1, ..., d . We may assume that the sets Ai are linearly ordered as in X. Let B i< denote the set Bi with covectors linearly ordered as in X. If n j1 is the number of covectors between the position of C j in  Ai and \(C j ) in  B i< , and n j2 the number of covectors between the position of \(C j ) in | Ai and C j in | B i< then n j1 +n j2 is equal to the total number of covectors satisfying C j ) in X. Setting n j =n j1 +n j2 , and summing over [C j ] j=1, ..., d ,  dj=1 n j =d&2. Thus in d&2 transpositions we may

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simultaneously reorder each  B i< as  Bi , and each | B i< as an extensor  Di satisfying 1: | Ai and  Di are identical except in set of common positions which are occupied by elements of [C j ] j=1, ..., d . 2) If a position of | Ai contains \(C j ) then the corresponding position of  Di contains C j . In this case sgn

\

d

+

 | 7 Ai _sgn i=1

\

d

+

 7 Di =(&1) d&1 i=1

(24)

By an application of the dual of Lemma 5.6,

\

sgn

d

+

 | 7 Ai _sgn i=1

=sgn

\

d

\

d

 7 Ai i=1

+

 | 7 B i< _sgn i=1

\

+ d

 7 B i< i=1

+

(25)

Now, sgn

\

d

+

d

\ + sgn  7 D _sgn  | 7 B \ + \ +

 7 Bi _sgn

 7 B i<

i=1

i=1 d

=(&1) d&2

d

< i

i

i=1

(26)

i=1

From which it follows that sgn

\

d

+

 7 Ai _sgn i=1

\

d

+

 7 Bi =(&1) 2d&3 i=1

for a sign change of &1, as d2. Lemma 5.6. Let S i /a, 1ik be proper subsets of ordered vectors of a unimodular basis a=[a 1 , ..., a n ] in GC(n). Let  S i denote the join of vectors in S i , and | S i the supplement of S i . If  ki=1 6 S i =[a 1 , ..., a n ] k&1 =1, then sgn

_}  S , ..., }  S & =(&1) 1

p

k

where p is the constant 12(n 2 & ki=1(n& |S i | ) 2 ). Proof. If  ki=1 6 S i =[a 1 , ..., a n ] k&1, then the sets [ |S i ] i=1, ..., k partition a. Let the extensor |S j be denoted as a j1 6 } } } 6 a jn& |Sj | , where

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MICHAEL HAWRYLYCZ

each j i # [1, ..., n]. We apply Hodge duality with respect to a unimodular basis. Applying the Hodge* operator to the join of these vectors, Va j1 6 } } } 6 a j n& |S j | =(&1) j1 + } } } +j(n& |S j | )&card( |S j )(card( |S j )+1)2 a p1 6 } } } 6 a p |S j | where a p are the vectors of a linearly independent from a j . By Proposition 2.9 the operator * is an isomorphism of (G(V), ) and (G(V), ) so, sgn

\}

}

+

 S 1 , ...,  S k =sgn

\

=sgn V

k

 a i 1 } } } a i n&|S i | i=1

\

+

k

+ +

 a i 1 } } } a i n& |S i | i=1

k

\ =sgn \ \n & : (n& |S |) ++ sgn \  6 S + =sgn

 Va i1 } } } a i n& |S i |

i=1

k

1 2

k

2

2

i

i=1

i

i=1

where the last equality holds as  ki=1 card( |S i )=n.

6. ENLARGEMENT OF ARGUESIAN IDENTITIES In this section we prove a dimension independence result for Arguesian E polynomials. Theorem 6.1 effectively states that P # Q is an identity of Arguesian polynomials iff each identity in an infinite set of extensions of the original is valid. The higher dimensional identities are constructed by a formal substitution of variables. Let P be a type I Arguesian polynomial in GC(n) on vector set a and covector set X. We define the enlargement by k of P(a, X) to be the multilinear Arguesian polynomial P (k)(a (k), X (k) ) in step GC(n) on variable set a (k) =[a i, l | i=1, ..., n, l=1, ..., k] and X (k) =[X j, m | j=1, ..., n, l=1, ..., k] in which each vector a i # a is formally replaced by the join of distinct vectors a i, 1 6 a i, 2 6 } } } 6 a i, k and each repeated covector X jp # X* of label X j , is replaced by meet of distinct covectors X jp, 1 7 X jp, 2 7 } } } 7 X jp, k , where X jp, l # X (k)*. The enlargement of a type II Arguesian polynomial is defined analogously. The variable sets a (k) (and X (k) ) are ordered by convention as, for a i, l , a i $, l $ # a (k), a i, l
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Theorem 6.1 (Enlargement). Let P(a, X) and Q(a, X) be non-zero Arguesian polynomials of either type in GC(n). Let P (k)(a (k), X (k) ) and Q (k)(a (k), X (k) ) be enlargements by k of P and Q in GC(k } n). Then E

E

P # Q  P (k) # Q (k). For example, the identity for Desargues' Theorem is [a, b, c](a 6 BC) 7 (b 6 AC) 7 (c 6 AB) E

# [[A, B, C]](bc 7 A) 6 (ac 7 B) 6 (ab 7 c) By applying Theorem 6.1 we obtain the identity (27) and geometric interpretation of Theorem 6.2. This identity is also valid by Theorem 4.1 and is a consequence of the Arguesian lattice law. Theorem 6.2. In 5-dimensional projective space let l 1 , l 2 , l 3 and l $1 , l $2 , l $3 be lines. Then the solids spanned by the pairs [l 1 , l $1 ], [l 2 , l $2 ], [l 3 , l $3 ] contain a common point iff the line formed by intersecting the solids l 1 l 2 & l$1 l $2 , the line formed by intersecting the solids l 2 l 3 & l $2 l$3 , and the line formed by intersecting the solids l 1 l 3 & l $1 l$3 all lie in a common 4-dimensional hyperplane. As a GC(6) identity, setting a (2) =a 1 a 2 , A (2) =A 1 A 2 , [a (2), b (2), c (2) ](a (2) 6 B (2)C (2) ) 7 (b (2) 6 A (2)C (2) ) 7 (c (2) 6 A (2)B (2) ) E

# [A (2), B (2), C (2) ](b (2)c (2) 7 A (2) ) 6 (a (2)c (2) 7 B (2) ) 6 (a (2)b (2) 7 C (2) ). (27) The following theorem and Arguesian identity is not a direct consequence of Theorem 4.1. Nevertheless Theorem 6.1 guarantees that its enlargement is valid. Theorem 6.3. In five-dimensional projective space, the lines ab$c$e$ & bb$d $f $

cb$c$d $ & da$e$f $

ea$d $f $ & fa$c$e$

lie on a common four-dimensional hyperplane, iff the three solids formed by the span of the lines [abc & b$c$d $e$f $, def & a$c$d $e$f $], the span of the lines [bde & a$b$d$e$f $, adf & a$b$c$e$f $] and the span of the lines [bce & a$b$c$d$f $, acf & a$b$c$d $e$] contain a common point. [a, b, c, d, e, f ] 2 ((a 6 ADF) 7 (b 6 ACE)) 6 ((c 6 AEF) 7 (d 6 BCD)) 6 ((e 6 BCE) 7 ( f 6 BDF)) E

# [[A, B, C, D, E, F]] 2((abc 7 A) 6 (def 7 B)) 7 (bde 7 C) 6 (adf 7 D)) 7 ((bce 7 E) 6 (acf 7 F ))

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(28)

38

MICHAEL HAWRYLYCZ

By Theorem 6.1 enlarge identity (28) to the following identity valid in GC(18). We leave the interpretation of this identity ot the reader. [a (3), b (3), c (3), d (3), e (3), f (3) ] 2(((a (3) 6 A (3)D (3)F (3) ) 7 (b (3) 6 A (3)C (3)E (3) )) 6 ((c (3) 6 A (3)E (3)F (3) ) 7 (d (3) 6 B (3)C (3)D (3) )) 6 ((e (3) 6 B (3)C (3)E (3) ) 7 ( f (3) 6 B (3)D (3)F (3) )) E

# [[A (3), B (3), C (3), D (3)E (3), F (3) ]] 2((a (3)b (3)c (3) 7 a (3) ) 6 (d (3)e (3)f (3) 7 B (3) )) 7 ((b (3)d (3)e (3) 7 C (3) ) 6 (a (3)d (3)f (3) 7 D (3) )) 7 ((b (3)c (3)e (3) 7 E (3) ) 6 (a (3)c (3)f (3) 7 F (3) ))

(29)

Lemma 6.4. Let P(a, X) be a non-zera type I Arguesian polynomial and P (k)(a (k), X (k) ) an enlargement, with ? (k)* a pre-transversal of P (k)*. Then k

? (k)*= . ? p(k)* p=1 (k)

(k)

where for each p=1, ..., k, ? *: a  X (k)* is a partial mapping such that each ?* p /a_X*, obtained by deleting second subscripts from the variables , is a pre-transversal ?*: a  X* of P*. Further, for any Q (k) P (k), of ? p(k)* E (Q (k) )| ?(k) is non-zero if and only if for each p=1, ..., k, E (Q)| ?p is non-zero for corresponding QP. Proof. An elementary induction shows that [a i, l , X j1, m ] # E (P (k)*) iff [a i , X j1 ] # E (P*) for all l, m=1, 2, ..., k, X j1 # X*, X j1, m # X (k)*, a # a, a i, l # a (k). Let B be the bipartite multigraph to associated to P and B (k) be the bipartite multigraph associated to P (k). From the previous remarks, the multigraph B (k) may be constructed from B by replacing each vertex with label a i # a, (or X j # X) of B by k distinct vertices a i, l # a (k), (and X j, m # X (k) ), l, m=1, ..., k in B k. Two distinct vertices a i, l and X j, m of B k are connected with an edge (a i, l , X j, m ) if and only the associated vertices a i and X j have edge (a i , X j ) in B. For any l, m=1, ..., k we shall say that the edge (a i, l , X j, m ) of B k is associated with the edge (a i , X j ) of B. Let M (k) be a perfect matching of B (k). The edges of M (k) may be partitioned into disjoint M (k) = kp=1 M p such that for each M p , the associated set of edges to M p in B forms a perfect matching of B. For given the induced subgraph M (k) of B k form the k-regular bipartite multigraph B M by contracting to a$i all vertices a i, l , l=1, ..., k in M (k) associated to a i and to X$j all associated vertices X j, m associated to X j . An application of the BirkhoffVonNeumann Theorem shows that B M may be factored into k-distinct perfect matchings. As a consequence, an edge (a i, l , X j, m ) of B (k)

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ARGUESIAN IDENTITIES IN INVARIANT THEORY

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is contained in a perfect matching of B (k) only if its associated edge (a i , X j ) is contained in a perfect matching of B. The converse is evident. Given a pre-transversal ?* of Arguesian P(a, X), define its canonical extension ?^ (k)*, to be the pre-transversal of P (k)* defined as ? (k): a i, l  X*j, l (k) for l=1, ..., k, X* *, if ?: a i  X j* , where X * j, l # X j is associated to X* j, l . It is not difficult to see that a pre-transversal ?* of P* is a transversal ? of P iff ?^ (k)* is a transversal ?^ (k) of P (k). Given transversal ? with ?*: a i  X j* , a transversal ? (k) with ? (k)*: a i, l  X*j, m for any l, m=1, ..., k may be easily (k) (k) *: a i, m  X* constructed as follows: If ?^ (k)*: a i, l  X* j, l and ?^ j, m form ?^$ (k) (k) by setting ?^ *(a j, s )=?^$ *(a j, s ) for j=1, ..., n, j{i, s=1, ..., k, and for (k) * : a i, m  X*j, l . j=i, all s{l, m. Then set ?^$ (k)* : a i, l  X* j, m and ?^$ (k) k (k) (k) (k) ( O ) Suppose _? *= p=1 ? i * and Q P such that E (Q (k) | ?(k) ){0 but for some p # [1, ..., k] the associated pre-transversal ?* p of P* has E (Q)| ? p =0. The canonical extension ?^ p(k)* containing the partial matching =0. Therefore by Proposition 3.17, ?^p(k)* : a i, l  X*j, m satisfies E (Q (k) )| ?^ (k) p the Grassmann condition holds in Q (k) under ?^ (k) p . Case 1. _R (k) 7 S (k) Q (k) with [X j, 1 , ..., X j, k ] ext(E (R (k) ))| ?^p(k) & ), for some j # [1, ..., n]. Let M (k) denote the perfect matching ext(E (S (k) )| ?^ (k) p (k) in B corresponding to ?^ (k)* of P (k)* with M (k) = kp=1 M p where M p denotes the set of edges corresponding to ?^ p(k)* . The set M (k)"M p is the disjoint union of (k&1) sets, each of whose associated edges forms a perfect matching of B. The pre-transversal ? (k)* may now be reconstructed by removing the vector-covector assignments of ?^ l(k)* corresponding to M l # M (k)"M p and reassigning according to ? (k)*. If (a t, l , X q, m ) and (a t$, l $ , X q$, m$ ) are both edges of M l then t{t$ and q{q$. Therefore, in replacing each assignment at most one covector with label from the set [X j, m : m=1, ..., k] is reassigned for each p. Since only (k&1) sets M l are reassigned, the assignment of some X j, m # [X j, 1 , ..., X j, k ] must remain unchanged. By Lemma 3.13, X j, m # ext(E (R (k) )| ?(k) & ext(E (S (k) )| ?(k) ) so that E (R (k) )| ?(k) 7 E (S (k) )| ?(k) =0, a contradiction. Case 2. _R (k) 7 S (k) Q (k), and the extension ?^ p(k)* of pre-transversal (k) )| ?^ p(k) ) and ext(E (S (k) )| ?^ p(k) ) contain a set C of covec?* p , satisfies ext(E (R tors which do not span X (k). This case is similar. ( o ) Suppose _? (k)*= ki=1 ? i(k)* and Q (k) P (k) such that E (Q)| ?p is non-zero for each p=1, ..., k, yet E (Q (k) )| ?(k) =0. Then either Case 1. _R (k) 7 S (k) Q (k) with X j, m # ext(E(R (k) )| ?(k) ) & ext(E (S (k) )| ?(k) ) for some j # [1, ..., n], m # [1, ..., k]. If _a i, l # V(R (k) ) such that [a i, l , X j, m ] # E (R (k) )| ?(k) , then as X j, m # ext(E (R (k) )| ?(k) ), type I R (k) satisfies the hypothesis of Lemma 3.13 (part 2). Then for every pre-transversal _ (k)* of P (k)* with E (R (k) )| _(k) non-zero, [a i, l , X j, m ] # E (R (k) )| _(k)

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MICHAEL HAWRYLYCZ

and X j, m # ext(E (R (k) )| _(k) ). Hence there is no a i $, l $ # V(S (k) ) with [a i $, l $ ,X j, m ] # E (S (k) )| _(k) ), and as X j, m # ext(E (S (k) )| ?(k) ), by Lemma 3.13 (part 1), X j, m # ext(E (S (k) )| _(k) ). Therefore X j, m # ext(E (R (k) )| _(k) ) & ext(E (S (k) )| _(k) ), for the arbitrary pre-transversal _ (k)* and P (k) is zero, a contradiction. Thus R (k), S (k) P (k) both satisfy Lemma 3.13 (part 1). A straightforward induction shows that the corresponding R, SP are type I satisfying Lemma 3.13 (part 1) as well. That is, R/P satisfies: For any pre-transversal ?* of P* with E (R)| ? non-zero, [a i , X j ] # E (R)| ? for some a i # V(R) iff X j  ext(E (R)| ? ). By hypothesis, E (Q)| ?p is non-zero, for p=1, ..., k. Hence 7 S)*), each of label there exists a set of repeated covectors [X *]/C((R j X j , such that for p=1, ..., k, ? p* : a i  X j* for some a i # V(R 7 S) and some (k) X j* # [X j*]. As ? * i are obtained by deleting second subscripts from ? i * , (k) (k) (k) pre-transversal ? * maps some vector of V(R 7 S ) to a repeated (k) : a (k)  X (k) is a covector of the set [X* j, 1 , ..., X* j, k ]. As the projection ? (k) (k) (k) (k) bijection, the image of V(R 7 S ) under ? | R(k) , ? | S(k) contains the entire set of labels [X j, 1 , ..., X j, k ]. Then for X j, m , _a i, l # V(R (k) 7 S (k) ) such that [a i, l , X j, m ] # E (R)| ?(k) or [a i, l , X j, m ] # E (S )| ?(k) , and X j, m  ext(E (R (k) )| ?(k) ) & ext(E (S )| ?(k) ), a contradiction. Case 2 is similar. K Proof of the Enlargement Theorem. Let P be type I and Q type II Arguesian, although the proof is valid if P, Q have the same type. Let ? (k) be a transversal of P (k) with factorization ? (k)*= kp=1 ? p(k)*. By Lemma 6.4 each partial mapping ? p(k)*, for p=1, ..., k, has associated E (P)| ? p non(k) *, the zero, i.e. ? p is a transversal of P. If ? p(k)* : a i, l  X* j, m in P E associated transversal of P satisfies ? p* : a i  X * j . As P # Q, regarding ? p as a transversal of Q, the canonical extension ?^ (k) in Q (k) identifies by p (k) * such that [a i, l , X j, m ] # E (Q (k) ). As Q (k) Lemma 3.20 the unique a* i, l # a is multilinear in covectors, setting ? (k)*: X j, m  a*i, l , the partial mappings ?p(k)* , p=1, ..., k, and pre-transversal ? (k)* are well-defined in Q (k)*. Then E E (Q)| ?p is non-zero for each partial mapping ? p(k)* as P # Q, so by the dual of Lemma 6.4, E (Q)| ?(k) is non-zero, or ? (k) is a transversal of Q (k). The coefficient of each transversal in E (P (k) )| ?(k) is \1 by Proposition E 3.15. To show that P (k) # Q (k) it remains to show that sgn(P (k) | ?(k) )= sgn(Q (k) | ?(k) ) for every transversal ? (k) or sgn(P (k) | ?(k) )=&sgn(Q (k) | ?(k) ) for every transversal ? (k). We may relate the sign of a transversal of P (k) to the sign of a transversal of P by the following steps: 1. Given a transversal ? of non-zero P with ?: a i  X j , calculate the sign of the canonical extension ?^ (k) of P (k), ?^ (k): a i, l  X j, l for l=1, ..., k, as sgn(E (P)| ?^(k) )=(&1) k sgn(E (P)| k ). 2. A transversal _ (k) of P (k) represents a matching, M (k) = (k) M s in k B in which each set M p corresponds to a partial mapping _ (k) p which determines a transversal _ p of P.

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3. Apply a set of transpositions converting _ (k) =[_ (k) p ] p=1, ..., k to a new transversal _^ (k) =[_^ p(k) ] p=1, ..., k such that for all p=1, ..., k, _^ (k) p assigns remains a a i, p to a covector X j, p with second subscript p, and each _^ (k) p partial mapping of P (k) determining a transversal _^ p of P. By Lemma 7.1 each transposition reverses the sign of the previous transversal. (k) (k) (k) such that for 4. Form a sequence _^ (k) =\ (k) 0 , \ 1 , ..., \ k =?^ (k) (k) * except for vectors a i, p for p=1, ..., k, \ p is a transversal, and \ p *=\ p&1 reassign which _^ p(k) : a i, p  X j, p in which case if ?^ (k)*: a i, p  X* j $, p \ p(k)* : a i, p  X* j $, p . Then (k) )=sgn(E (P) sgn(E (P (k) )| \ (k) )_sgn(E (P (k) )| \ p&1 _p )_sgn(E (P)| ? ) p

Steps 1 and 3 are straightforward and Step 2 is proven in Lemma 6.4. To prove the sign equality of step 4 we require two final lemmas whose proofs are not difficult and are omitted. The lemmas remains valid when  and  are interchanged, and vectors and interchanged with covectors. Lemma 6.5. Let Ai , i=1, ..., 4 each be ordered sets of covectors with C(Ai )X for i=1, ..., 4 with |A1 | = |A3 |, and |A2 | = |A4 |. Let A (k) i , (k) | = |A i=1, ..., 4 be ordered sets of covectors C(A (k) )X (k) with |A (k) 1 3 | (k) | = |A |. and |A (k) 2 4 (k) (k) (resp. A (k) Suppose that the ordered sets A (k) 1 2 ) and A 3 (resp. A 4 ) are identical except in a set S(resp. T ) of common positions which are occupied by covectors X j, s , having second subscript s, for some 1sk. Suppose also that Ai , for i=1, ..., 4, is the ordered set of associated covectors of the subset of A (k) i , with second subscript s. Then

\

+

\

+ =sgn  A 6  A _sgn  A 6  A \ + \ +

(k) (k) _sgn  A (k) sgn  A (k) 1 6  A2 3 6  A4

1

2

3

4

(30)

Lemma 6.6. Let A1 , A2 be ordered sets of distinct vectors V(Ai )a, (k) i=1, 2 with |V(A1 )| = |V(A2 )|, and let A (k) 1 , A 2 be ordered sets of distinct (k) (k) (k) (k) vectors V(A i )a , with |V(A 1 )| = |V(A (k) 2 )|, Suppose that A i , i=1, 2 are identical except in a set S of common positions which are occupied by vectors [a l, s ] with second subscript s for some 1sk. Suppose also that the associated set to the ordered subset of A (k) i , deter(k) be mined by positions S is precisely Ai , i=1, 2. Let B1 , B2 , B (k) 1 , B2 ordered sets of covectors satisfying the same conditions with covectors X m, s , 1sk and a set of positions T. (k) (k) Let R i =( Ai ) 6 ( Bi ) and R (k) i =( A i ) 6 ( B i ) be type I exten(k) (k) : A  B , i=1, 2 assignments from vectors to sors with ? i : Ai  Bi , ? (k) i i i

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(k) covectors such that ? (k) except on vectors and covectors with second 1 =? 2 subscript s, which are assigned to each other such that ? (k) i : a l, s  X m, s iff ? i : a l  X m for i=1, 2. Then (k) (k) (k) sgn(E (R (k) 1 )| ? 1 )_sgn(E (R 2 )| ?2 )=sgn(E (R)| ?1 )_sgn(E (R)| ?2 )

(31)

To complete the proof of step 4, we show recursively that for any p=1, ..., k, and type I Q (k) P (k) with corresponding QP. (k) )=sgn(Q | sgn(Q (k) | \ p(k) )_sgn(Q (k) | \ p&1 _p )_sgn(Q | ? )

(32)

Any non-trivial type I Arguesian P, contains Q=e=a 1 } } } a l  X 1 } } } X m , and thus P (k) contains Q (k) =e (k) =a 1, 1 } } } a l, k  X 1, 1 } } } X m, k . Setting (k) A1 =  A2 = a 1 } } } a l ,  B1 =  B2 = X 1 } } } X m , A (k) 1 = A 2 = (k) (k) (k) a 1, 1 } } } a l, k 7 B 1 = B 2 =X 1, 1 } } } X l, k , we have that \ p and \ (k) p&1 (k) satisfy the hypothesis of ? (k) of Lemma 6.6 (or its dual) with respect 1 , ?2 (k) )=sgn(E (e)| to _ p and ?. Thus sgn(E (e (k) )| \p(k) )_sgn(E (e (k) )| \ p&1 _p )_ sgn(E (e)| ? ). The covectors may be reordered, without global change ) and of sign; such that X j, l occurs identically in ext(E (e (k) )| \ (k) p (k) ) in position i if l{p and if l=p and X appears in posiext(E (e (k) )| \ p&1 j, p ) tion i of ext(E (e (k) )| \ p(k) ) then Y m, p appears in position i of ext(E (e (k) )| \ (k) p&1 where associated X j and Y m occur in identical positions of ext(E (e)| ? ) and ext(E (e)| _p ). Inductively, there are three cases: Case 1. Q (k) =R (k) 6 S (k) where both R (k) and S (k) are type I, and ), by induction the result holds for R (k) and S (k) and ext(E (R (k) )| \ (k) p (k) ) satisfy the above reordering property with respect to ext(E (R (k) )| \ p&1 ext(E (R)| _p ) and ext(E (R)| ? ) and similarly for S (k) and S. Setting (k) (k) (k) )= )| \ p(k) )= A (k) )| \ p&1 ext(E (R (k) )| \ p(k) )= A (k) 1 , ext(E (S 2 , ext(E (R (k) (k) (k) )=A while ext(E (R)| )= A , ext(E (R)|  A 3 , ext(E (S )| \ (k) ? 1 _p ) 4 p&1 = A3 , ext(E (S)| _p )= A4 , the hypothesis of Lemma 6.5 are satisfied. Then sgn(E (Q (k) )| \ (k) )_sgn(E (Q (k) )| \ (k) ) p&1 p =sgn(E (R

(k)

6S

(k)

| \ p(k) )_sgn((E (R

(33) (k)

6S

(k)

)| \ (k) ), p&1

(34)

=sgn(E (R (k) )| \ (k) )_sgn(E (S (k) )| \ p(k) ) p _sgn(E (R (k) )| \ (k) 6 E (S (k) )| \ (k) ) p p _sgn(E (R (k) )| \ (k) )_sgn(E (S (k) )| \ (k) ) p&1 p&1 (k) ). _sgn(E(R (k) )| \ (k) 6 E (S (k) )| \ p&1 p&1

(35)

By induction hypothesis, (k) )=sgn(E (R)| sgn(E (R (k) )| \ p(k) )_sgn(E (R (k) )| \ p&1 _ p )_sgn(E (R)| ? )

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(36)

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43

and similarly for S, S (k). By Lemma 6.5 equation (35) may be written as sgn(E (R)| _p )_sgn(E (R)| ? )_sgn(E (S)| _p )_sgn(E (S)| ? ) _sgn(E (R)| _p 6 E (S)| _p )_sgn(E (R)| ? 6 E (S)| ? )

(37)

=sgn(E (R 6 S)| _p )_sgn(E (R 6 S)| ? )=sgn(E (Q)| _p )_sgn(E (Q)| ? ) (38) Hence (32) is satisfied, and the reordering property holds on the covectors ), with respect to ext(E (Q)| _p ), of ext(E (Q (k) )| \ p(k) ) and ext(E (Q (k) )| \ (k) p&1 ext(E (Q)| ? ). The other cases are easily verified. E To complete the proof, as P # Q we may assume without loss of generality that for every transversal ?, sgn(E (P)| ? )=sgn(E (Q)| ? ). Any transversal _ (k) of P (k) and Q (k) may be converted to _^ (k) by steps 2 and 3. Since, by Lemma 7.1, each transposition is sign reversing in any Arguesian polynomial, and sgn(E (P (k) )| _(k) )_sgn(E (Q (k) )| _(k) )=sgn(E (P (k) | _^(k) )_sgn(E (Q (k) | _^(k) ) (39) (k) (k) (k) is a sequence of transversals The sequence _^ (k) =\ (k) 0 , \ 1 , ..., \ k =?^ such that for p=1, ..., k, (k) ) )_sgn(E (P (k) )| \ p&1 sgn(E (P (k) )| \ (k) p

=sgn(E (P)| _p )_sgn(E (P)| ? ) =sgn(E (Q)| _p )_sgn(E (Q)| ? ) (k) ) )_sgn(E (Q (k) )| \ p&1 =sgn(E (Q (k) )| \ (k) p

(40)

E

as P # Q, applying step 4, and regarding _ (k) as a transversal of Q (k). By repeated application of (40), sgn(E (P (k) )| _^(k) )_sgn(E (P (k) )| ?^(k) )= sgn(E (Q (k) )| _^(k) )_sgn(E (Q (k) )| ?^(k) ) and then by (39), sgn(E (P (k) )| _(k) )_sgn(E (P (k) )| ?^(k) ) =sgn(E (Q (k) )| _(k) )_sgn(E (Q (k) )| ?^(k) )

(41)

Given any ? of P, the canonical ?^ (k) satisfies sgn(E (P (k) )| ?^(k) )=(&1) k sgn(E (P)| ? ). Then by step 1, sgn(E (P (k) )| _(k) )_sgn(E (Q (k) )| _(k) ) =(&1) k sgn(E (P)| ? )_(&1) k sgn(E (Q)| ? )=1, Theorem 6.1 is proved. K

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7. THE TRANSPOSITION LEMMA In his study of Cayley factorization, White [19] has studied the change of sign upon permuting variables in GrassmannCayley algebra expressions. Lemma 7.1 shows that in the case of Arguesian polynomials, if the permutation \ induced by transversals ?, _ is a transpositions then sgn(E (P)| ? )_ sgn(E (P)| _ )= &1. In general, given transversals ?, _ of P there does not necessarily exist a sequence of transversals ?=? 0 , ..., ? s =_ with \ i induced by ? i , ? i+1 0in&1 a transposition. Further, Lemma 7.1 is somewhat surprising in that no obvious extensions based on standard permutation statistics, such as cycle length, or number of inversions of \ are valid. Lemma 7.1. Let P be an Arguesian polynomial in a GC(n) with transversals ? and _ such that ?=_ except ?: a i  X l , ?: a j  X m while _: a i  X m , _: a j  X l for distinct a i , a j # a, X l , X m # X. Then sgn(E (P)| ? )=&sgn(E (P)| _ ) Proof. We show that if ? and _ differ by a transposition, then only four possible expansions in E*(P) are possible. Then we verify that in each case the resulting transverals differ in sign. The proof depends strongly on the assumption of multilinearity of one of the variable sets. The expansions are 1. R*(a i , a j )  S*(X * l , X* m ), 2. (R*(a i , a j ) 7 S*(X *))/R$*(a l i , a j ), and R$*(a i , a j ) S$*(X* m ), 3. (R*(a i ) 6 S*(X l* , X* m ))/S$*(X * l , X* m ), R$*(a j ),

and

S$*(X l* , X* m ) 

4. _ two type I R 1*(a i ) 6 S 2*(X l1 , X m2 ), R 2*(a j ) 6 S 2*(X l2 , X m1 ). where X * l , X* m are the unique covectors satisfying the hypothesis of the Lemma. Let P be type I and by Lemma 3.20 let unique X l1 , X l2 , X m1 , X m2 # X* satisfy ?*: a i  X l1 , a j  X m1 , _*: a i  X m2 , a j  X l2 , where possibly l 1 =l 2 or m 1 =m 2 . As [a i , X l1 ] # E (P*)| ?* there is either type III Q=R  S, with a # V(R), X l1 # C(S*) and linear combinations R*(a i ), S*(X l1 ). First suppose for type I S/P, X l1 , X l2 # C(S*) (equivalently X m1 , X m2 # C(S*)) with both a i , a j  V(S). Then ?*: a i  X l1 and _*: a j  X l2 necessarily imply X l1 # ext(E (S*)| ?* ), X l2 # ext(E (S*)| _* ), and by Lemma 3.19, for every transversal #, [a, X l ] # E (S)| # , for some a # V(S), which is a contradiction. It follows at once that if Q=R  S, with ai , a j # V(R), and X l1 , X l2 , X m1 , X m2 # C(S*), then l 1 =l 2 , m 1 =m 2 . Denoting

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these unique covectors as X * l , X* m , the expansion E (Q*) is given as case 1 in the list above. (a) Now suppose Q=R 7 S is type II, with R type II, S type I, and a i # V(R), X l1 # C(S*). By Lemma 3.13 (part 3), the covector X l # X satisfies [a, X l ] # E (Q)| # for a # V(Q), for every transversal # of P. Thus if a j  V(S) then a j # V(R). Again by Lemma 3.19 l 1 =l 2 and we may write for unique X * R*(a i , a j ) and S*(X *) l l of label X l . If R 7 S is minimal with respect to (a) then we may further assume that Xm1 , Xm2  C(R*) for suppose X m1 # C(R*). As ?*: a j  X m1 , _R$  S$R with type II R$, type I S$, a j # V(R$), X m1 # C(S$*). Clearly we may not have type I R$ 6 S$R, as R*(a j ). Therefore R$ 7 S$R, and a i  V(S$) or R*(a i ) is violated. Then a i # V(R$), and R$ 7 S$R is a smaller subexpression satisfying (a). (b) On the other hand, if Q=R 7 S with a i # V(R), X l1 # C(S*) but a j # V(S). Then there exists type I Q$=R$ 6 S$S with a j # V(R$), and X l2 # C(S$*). Then X m1 # C(S$*), X l2 # C(S$*), and S$*(X l2 , X m1 ) is required. If minimal (a) occurs but case 1 does not, R 7 S/P with R*(a i , a j ), and R/R$ with R$  S$P minimal such that R$*(a i , a j ), and S*(X *), l S$*(X m1 ) or S$*(X m2 ). If either  then X m1 # ext(E (S$*)| ?* ) and X m2 # ext(E (S$*)| _* ) which is a contradiction. Thus again m 1 =m 2 and denoting the unique covectors as X l* , X*m , we obtain case 2 above. If (b) occurs but (a) does not occur, then consequently cases 1 or 2 do not occur. We may assume that Q=R 6 S with a i # V(R), X l1 , X m2 # C(S*), and a j  V(Q). As _*: a j  X l2 , _Q$=R$ S$/P with R$*(a j ), S*$(X l2 ). As Q, Q$ are subexpressions of a parenthesized binary expression in , either QQ$, Q$Q or Q*, Q$* contain no common variables. If Q*, Q$* have no variables in common, and type I Q$=R$ 7 S$P then by Lemma 3.13 (part 3), ?*: a i  X l1 is violated. Hence R$ 6 S$P with R$*(a j ) and then S$*(X l2 , X m1 ), which is case 4 above. Otherwise, Q*, Q*$ share variables of a _ X* but a j  V(Q). As a j # V(R$), R$  3 Q. We may not have QR$ either as R$ is type II and _: a j  X l2 is violated. Thus R$* and Q* have disjoint variables. Then if S$* and Q* have variables in common and in particular, if S$/Q with inclusion proper, then R$ 6  7 S$ is not a subexpression. The only remaining possibility is that Q$=R$ 6  7 S$P with type I Q=R 6 SS$, and a j # V(R$). Then S*$(X l2 , X m1 ) is required, and we proceed to consider this case. Let Q=R 6 S with R*(a i ), S*(X l1 , X m2 ), a j  V(Q), and further suppose Q/S$ with S$  R$/P, and S$*(X l2 , X m1 ), R$*(a j ). Again we claim l 1 =l 2 , m 1 =m 2 . To show this we proceed as follows: [a i , X l1 ] # E (Q*)| ?* and if X l # ext(E (Q)| ? ), then by Lemma 3.13 (part 2), for every transversal #, [a, X l ] # E (Q)| # for a # V(Q). As a j  V(Q) this contradicts _*: a j  X l2 .

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Therefore Q satisfies the hypothesis of Lemma 3.13 (part 1) for X l . In fact, by Lemma 3.19, for any transversal # we must have, for any a # V(Q), [a, X l ]  E (Q)| # O X l1 # ext(E (Q*)| #* )

(43)

[a, X m ]  E (Q)| # O X m1 # ext(E (Q*)| _* )

(44)

We recursively evaluate the subexpressions T with QTS$ showing in fact that Eqs. (43) and (44) are valid when Q, Q* are substituted by substituted by T, T *. As P is non-zero, it follows that l 1 =l 2 , m 1 =m 2 as claimed. If any T is type II, Lemma 3.13 (part 3), contradicts ?*: a j  X m1 . Also, every type I T must satisfy Lemma 3.13 (part 1). By induction we obtain the contradiction that X l1 # ext(E (S$*)| _* ) and X l2 # ext(E (S$*)| _* ), and thus l 1 =l 2 , (m 1 =m 2 ) as required. This case gives the third of the list. It remains to show that sgn(E (P)| ? ) and sgn(E (P)| _ ) alternate. The sign of a transversal ? of a type I Arguesian polynomial is calculated recursively by Proposition 3.3. We shall verify only case 4, and may therefore assume that P contains two type I subexpressions Q 1 , Q 2 as in case 4. Since ?(a 1 )=_(a l ), l{i, j, we may write, E (Q 1 )| ? =k 1 k 2 X 1 } } } X m } } } X p "[X ?(1) } } } X l } } } X ?(k) ]

(45)

_sgn(X 1 } } } X m } } } X p "[X ?(1) } } } X l } } } X ?(k) ], X ?(1) } } } X l } } } X ?(k) )

(46)

where k 1 , k 2 are constants containing any brackets [a, X]. We may similarly write E (Q 1 )| _ =k 1 k 2 X 1 } } } X l } } } X p "[X _(1) } } } X m } } } X _(k)

(47)

_sgn(X 1 } } } X l } } } X p "[X _(1) } } } X m } } } X _(k) ], X _(1) } } } X m } } } X _(k) )

(48)

where Eqs. (45) and (47) are identical except for X l and X m occuring in possibly distinct positions. The meet X l } } } X m } } } X p "[X ?(1) } } } X l } } } X ?(k) ] occurs in ext(E (Q 1 )| ? ), and in the corresponding sign term. Hence sgn(E (Q 1 )| ? ) is unaffected by simultaneously transposing X m within (45) and within the sign of (46), to the position occupied by X l in the corresponding term of (47). The positions of X l and X m are identical in X ?(1) } } } X l } } } X ?(k) and X _(1) } } } X m } } } X _(k) since the position of a i is fixed. Now (45) and (47) differ only in that X l of (45) is exchanged for X m in (47), while (46), (48) are identical except for a transposition of X l and X m . The same holds for E (Q 2 )| ? and E (Q 2 )| _ . Equations (46), (48) differ differ by a sign change as do the corresponding sign terms of E (Q 2 )| ? and

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E (Q 2 )| _ . Both may be ignored since they will not contribute to the overall change in sign of sgn(E (P)| ? )_ sgn(E (P)| _ ). As P is type I _ type I R containing both Q 1 and Q 2 . Evaluate E (R)| ? and E (R)| _ recursively. Any T $Q 1 has E (T$)| ? =E (T $)| _ . Suppose first that R=Q 1 7 Q 2 identically. Then E (R)| ? =k 1 X i1 } } } X ip 7 X m 7 X ip+1 } } } X iq 7 X j1 } } } X jr 7 X l 7 X jr+1 } } } X js (49) E (R)| _ =k 1 X i1 } } } X ip 7 X l 7 X ip+1 } } } X iq 7 X j1 } } } X jr 7 X m 7 X jr+1 } } } X js (50) and after sign adjustment, (49) and (50) differ by a transposition alone. Hence interchanging X l , X m every subexpression RTP evaluates identically and sgn(E (P)| ? = &sgn(E (Q)| _ ). The case R=Q 1 6 Q 2 is similar. If the subexpression R containing Q 1 and Q 2 is neither Q 1 7 Q 2 nor Q 1 6 Q 2 , let Q 1 be innermost and let P contain (Q 1 7 S) 6 T, with possibly empty S or T, with E (S)| ? =E (S)| _ and E (T )| ? =E (T )| _ . We may not have X l or X m in ext(E (S)| ? )( =E (S)| _ ) since ? and _ are both non-zero. For the same reason ext(E (T )| ? ) must contain both X l and X m in identical positions. The evaluations (E (Q 1 )| ? 7 E (S)| ? ) 6 E (T )| ? and (E (Q 1 )| _ 7 E (S)| _ ) 6 E (T )| _ may be assumed to have the form, X i1 } } } X ip 7 X m 7 X ip+1 } } } X iq 7 E (S)| ? ) 6 X m1 } } } X ms 7 X l 7 X m

(51)

X i1 } } } X ip 7 X l 7 X ip+1 } } } X iq 7 E (S)| _ ) 6 X m1 } } } X ms 7 X l 7 X m

(52)

Setting B=X"[X i1 , ..., X m , ..., X iq ], B$=X"[X i1 , ..., X l , ..., X iq ] and splitting the extensor on the right (51) and (52) become [X i1 } } } X ip 7 X m 7 X ip+1 } } } X iq 7 E (S)| ? ), B"[X l ], X l ] _ X m1 } } } X ms 7 X l 7 X m "B _sgn(X m1 } } } X ms 7 X l 7 X m "B, B"[X l ], X l )

(53)

[X i1 } } } X ip 7 X l 7 X ip+1 } } } X iq 7 E (S)| _ ), B$"[X m ], X m ] _X m1 } } } X ms 7 X l 7 X m "B$] _sgn(X m1 } } } X ms 7 X l 7 X m "B$, B$"[X m ], X m )

(54)

In Eqs. (53) and (54), each bracket and its corresponding sign differ by a transposition of X l and X m alone, all other covectors being equal. Interchange X l and X m in (54) in both the extensor and sign term without affecting sgn(E(P)| ? )_ sgn(E (P)| _ ). The covectors of the extensors of (53) are identical to those of (54) with X l and X m of this new extensor occupying identical

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48

MICHAEL HAWRYLYCZ

positions. A simple induction shows that sgn(E (R)| ? )= &sgn(E (R)| _ ) so sgn(E (P)| ? )=&sgn(E (P)| _ ) as required. K

ACKNOWLEDGMENTS This research forms part of the author's doctoral thesis supervised by Professor Gian-Carlo Rota at MIT. I thank him for his guidance and encouragement.

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